Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

Series in Chemical Physics The Series in Chemical Physics is an international series that meets the need for up-to-date texts on theoretical and experimental aspects in this rapidly developing field. Books in the series range in level from introductory monographs and practical handbooks to more advanced expositions of current research. The books are written at a level suitable for senior undergraduates and graduate students, and will also be useful for practicing chemists, physicists, and chemical physicists who wish to refresh their knowledge of a particular field. Series Editors: J H Moore, University of Maryland, USA N D Spencer, ETH-Zurich, Switzerland series includes:

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications Jozef Kowalewski and Lena Mäler Fundamentals of Molecular Symmetry Philip R Bunker and Per Jensen

Series in Chemical Physics

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications Jozef Kowalewski Stockholm University, Sweden

Lena Mäler Stockholm University, Sweden

New York London

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Library of Congress Cataloging-in-Publication Data Kowalewski, Jozef. Nuclear spin relaxation in liquids : theory, experiments, and applications / by Jozef Kowalewski and Lena Mäler. p. cm. -- (Series in chemical physics ; 2) Includes bibliographical references and index. ISBN 0-7503-0964-4 1. Relaxation phenomena. 2. Nuclear spin. I. Mäler, Lena. II. Title. III. Series in chemical physics (Taylor & Francis Group) ; 2. QC173.4.R44K69 2006 530.4'16--dc22

2005054885

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Preface

Nuclear magnetic resonance (NMR) is a powerful branch of spectroscopy, with a wide range of applications within chemistry, physics, material science, biology, and medicine. The importance of the technique has been recognized by the Nobel prizes in chemistry awarded to Professor Richard Ernst in 1991 for methodological development of high-resolution NMR spectroscopy and to Professor Kurt Wüthrich in 2002 for his development of NMR techniques for structure determination of biological macromolecules in solution. NMR employs a quantum-mechanical property called spin, the origin of the characteristic magnetic properties of atomic nuclei, and its behavior in an external magnetic field. NMR provides discriminating detail at the molecular level regarding the chemical environment of individual atomic nuclei. Through-bond or through-space connections involving pairs of nuclei, rates of certain chemical reactions, and water distribution and mobility in tissue typify the information commonly available from the NMR experiment. Perhaps the richest information is obtained from studies of nuclear spin relaxation (NMR relaxation) in solution, which investigate how the transfer of energy and order occur among nuclear spins and between the spin system and its environment. Various aspects of NMR relaxation in liquids are the topic of this book. The relaxation phenomena are described in all major NMR books, with a varying level of depth and detail. We found it worthwhile to collect in a single volume the relaxation theory, experimental techniques, and illustrative applications. We go into more detail in these fields than is found in most general NMR texts. The goal is to explain relaxation in a way that would be possible for readers familiar with the basics of NMR to follow, such as advanced undergraduate and graduate students in chemistry, biochemistry, biophysics, and related fields. Most of the book is not too demanding mathematically, with the objective to illustrate and explain the physical nature of the phenomena rather than their intricate details. Some of the theoretical chapters (Chapter 4 through Chapter 6) go into more depth and contain more sophisticated mathematical tools. These chapters are directed more to specialists and may be omitted at first reading. Most of the key results obtained in these chapters are also given in the experimental or application sections. The outline of the book is as follows: After an introduction in Chapter 1, Chapter 2 and Chapter 3 present a fairly simple version of relaxation theory, including the description of dipolar relaxation in a two-spin system at the level of Solomon equations; they also contain a sketchy presentation of the intermolecular relaxation. Chapter 4 and Chapter 5 provide a more sophisticated

and general version of the theory, as well as a discussion of other relaxation mechanisms. Various theoretical tools are presented and applied to appropriate problems and examples. Chapter 6 deals with spectral densities, the link connecting the relaxation theory with the liquid dynamics. Chapter 7 is an introduction to experimental tools of NMR, with emphasis on relaxation-related experimental techniques for solution state studies. The following chapters cover experimental methods for studying increasingly sophisticated phenomena of single-spin relaxation (Chapter 8), cross-relaxation (Chapter 9), and more general multispin phenomena (Chapter 10). In each chapter, a progression of techniques is described, starting with simple methods for simple molecules and moving on to advanced tools for studies of biological macromolecules. Chapter 11 and Chapter 12 are devoted to applications of relaxation studies as a source of information on molecular dynamics and molecular structure. The remaining four chapters deal with a variety of special topics: • The effects of chemical (often conformational) exchange on relaxation phenomena and relaxation-related measurements (Chapter 13) • Relaxation processes in systems containing quadrupolar (I ≥ 1) nuclei (Chapter 14) • Paramagnetic systems containing unpaired electron spin (Chapter 15) • A brief presentation of NMR relaxation in other aggregation states of matter (Chapter 16) The book is not meant to be a comprehensive source of literature references on relaxation. We are rather selective in our choice of references, concentrating on reviews and original papers with pedagogic qualities that have been chosen at least partly according to our tastes. The references are collected after each chapter. Some of the NMR (and related) textbooks useful in the context of more than one chapter are collected in a special list provided immediately after this preface. Most of the figures have been prepared especially for this book; some are reproduced with permission of the copyright owners, who are acknowledged for their generosity. In the process of writing this book, we have obtained assistance from and support of several students and colleagues. Among these, we wish in particular to mention Hans Adolfsson, August Andersson, Peter Damberg, Jens Danielsson, Danuta Kruk, Arnold Maliniak, Giacomo Parigi, and Dick Sandström. We are very grateful to Nina Kowalewska and Hans Adolfsson for their endurance and support during the 2 years it took to write this text.

Further Reading

Abragam, A. The Principles of Nuclear Magnetism (Oxford: Oxford University Press), 1961. Atkins, P.W. and Friedman, R.S. Molecular Quantum Mechanics (Oxford: Oxford University Press), 1997. Bakhmutov, V.I. Practical NMR Relaxation for Chemists (Chichester: Wiley), 2004. Brink, D.M. and Satchler, G.R. Angular Momentum (Oxford: Clarendon Press), 1993. Canet, D. Nuclear Magnetic Resonance: Concepts and Methods (Chichester: Wiley), 1996. Cavanagh, J., Fairbrother, W.J., Palmer, A.G., and Skelton, N.J. Protein NMR Spectroscopy (San Diego: Academic Press), 1996. Ernst, R.R., Bodenhausen, G., and Wokaun, A. Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford: Clarendon Press), 1987. Goldman, M. Quantum Description of High-Resolution NMR in Liquids (Oxford: Clarendon Press), 1988. Hennel, J.W. and Klinowski, J. Fundamentals of Nuclear Magnetic Resonance (Harlow: Longman), 1993. Levitt, M.H. Spin Dynamics (Chichester: Wiley), 2001. McConnell, J. The Theory of Nuclear Magnetic Relaxation in Liquids (Cambridge: Cambridge University Press), 1987. Neuhaus, D. and Williamson, M.P. The Nuclear Overhauser Effect in Structural and Conformational Analysis (New York: VCH), 1989. Slichter, C.P. Principles of Magnetic Resonance (Berlin: Springer), 1989. Van Kampen, N.G. Stochastic Processes in Physics and Chemistry (Amsterdam: NorthHolland), 1981. Wüthrich, K. NMR of Proteins and Nucleic Acids (New York: Wiley), 1986.

The Authors

Jozef Kowalewski received his Ph.D. from Stockholm University in 1975. He began working with nuclear spin relaxation as a postdoctoral fellow at Florida State University. He became a professor in physical chemistry at Stockholm University in 1986 and chairman of the department of physical, inorganic, and structural chemistry in 1993. Dr. Kowalewski was a member of the chemistry committee of the Swedish Natural Science Research Council in the late 1990s and is currently a member of the advisory editorial board of Magnetic Resonance in Chemistry. He has supervised 15 Ph.D. students and authored about 150 scientific papers. Lena Mäler received her Ph.D. from Stockholm University in 1996, working with Jozef Kowalewski in nuclear spin relaxation. She was a postdoctoral fellow at the Scripps Research Institute and became an assistant professor in the Department of Biophysics in 1999. Dr. Mäler is currently associate professor at the Department of Biochemistry and Biophysics at Stockholm University.

Contents

1

Equilibrium and Nonequilibrium States in NMR ................................................................................. 1

2

Simple Relaxation Theory ........................................................... 19

3

Relaxation through Dipolar Interactions ................................... 39

4

The Redfield Relaxation Theory ................................................. 65

5

Applications of Redfield Theory to Systems of Spin 1/2 Nuclei ......................................................................... 97

6

Spectral Densities and Molecular Dynamics .......................... 127

7

NMR — the Toolbox ................................................................... 159

8

Measuring T 1 and T 2 Relaxation Times.................................... 177

9

Cross-Relaxation Measurements ............................................... 195

10 Cross-Correlation and Multiple-Quantum Relaxation Measurements .......................................................... 215

11

Relaxation and Molecular Dynamics ....................................... 243

12 Applications of Relaxation-Related Measurements to Structure Determination........................................................ 289

13 Relaxation and Chemical Exchange .......................................... 311 14 Effects of Quadrupolar Nuclei .................................................. 331 15 Nuclear Spin Relaxation in Paramagnetic Systems in Solution.................................................................... 359

16 Nuclear Spin Relaxation in Other Aggregation States ...................................................................... 385 Index ..................................................................................................... 411

1 Equilibrium and Nonequilibrium States in NMR

CONTENTS 1.1 Individual Spins: Elements of Quantum Mechanics ...............................2 1.2 Ensembles of Spins: the Density Operator................................................8 1.3 Simple NMR: the Magnetization Vector.................................................. 11 1.4 Coupled (Interacting) Spins: the Product Operator Formalism ....................................................................................16 References ............................................................................................... 18

The concepts of equilibrium and deviation from equilibrium have a central role in physical chemistry. Once a system is in a nonequilibrium state, it will tend to return to equilibrium in a process which does not occur instantaneously. This general phenomenon of development towards equilibrium is called relaxation. In the specific context of nuclear magnetic resonance, the equilibrium state is that of a macroscopic sample of nuclear spins in a magnetic field. In order to talk about a development towards equilibrium, we need to define and be able to create an initial state, which is different from the equilibrium state. Using experimental techniques of present day NMR spectroscopy — in the first place, radiofrequency pulses — it is possible to create a large variety of nonequilibrium states. Starting from a given nonequilibrium situation, the spin systems evolve in a complicated way in which the “return to equilibrium” processes compete with related phenomena, converting one type of nonequilibrium into another. In NMR, the processes by which the return to equilibrium is achieved are denoted “spin relaxation.” Relaxation experiments were among the earliest NMR applications of modern Fourier transform NMR, and the development of applications and experimental procedures has been enormous. Consequently, there is a vast literature on NMR relaxation regarding theoretical considerations and advances in methodology. Relaxation is important because it has effects on how NMR experiments are carried out, but perhaps even more valuable is the information content derived from relaxation 1

2

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

parameters. Information about the physical processes governing relaxation can be obtained from experimental NMR parameters. In order to set the stage for theoretical descriptions of spin relaxation, we need to define the spin system and the means of manipulating the spin systems to obtain nonequilibrium states. In this chapter, we provide elementary tools for description of spin systems and their equilibrium and nonequilibrium states.

1.1

Individual Spins: Elements of Quantum Mechanics

In order to arrive at a description of a large number of spins, we need to start with a single spin. The properties of spin are obtained using quantum mechanics. A brief account of the necessary quantum mechanical background is given in this section; see the further reading section in this book’s preface (in particular, the books by Atkins and Friedman, Goldman, Levitt, and Slichter) for a more complete presentation. The basis of quantum mechanics can be formulated as a series of postulates — fundamental statements assumed to be true without proof. The proof of these postulates lies in the fact that they successfully predict outcome of experiments. One field in which this works extremely well is nuclear magnetic resonance. The first postulate of quantum mechanics states that the state of a physical system is described, as fully as it is possible, by a wave function for the system. The wave function is a function of relevant coordinates and of time. In the case of spin, the wave functions are defined in terms of a spin coordinate. The important property of the spin wave functions is the fact that they can be formulated exactly in terms of linear combinations of a finite number of functions with known properties. The second postulate of quantum mechanics states that, for every measurable quantity (called observable), an associated operator exists. An operator, Qˆ , (we are going to denote operators, with certain exceptions specified later, with a “hat”) is a mathematical object that operates on a function and creates another function. In certain cases, the action of an operator on a function yields the same function, multiplied by a constant. The functions with this property are called the eigenfunctions of the operator and the corresponding constants the eigenvalues. In the context of spins, the most important operators are the spin angular momentum operators. Every nuclear species is characterized by a nuclear spin quantum number, I, which can attain integer or half-integer values. The nuclear spin quantum number is related to the eigenvalue of the total spin angular momentum operator, Iˆ 2 , through: Iˆ 2ψ = I ( I + 1)ψ where ψ is an eigenfunction of the operator Iˆ 2 .

(1.1)

Equilibrium and Nonequilibrium States in NMR

3

The third postulate of quantum mechanics states that if a system is described by a wave function that is an eigenfunction of a quantum mechanical operator, then the result of every measurement of the corresponding observable will be equal to that eigenvalue. Equation (1.1) assumes that the spin operators are dimensionless, a convention followed in this book. A natural unit for angular momentum in quantum mechanics is otherwise  , the Planck constant divided by 2π (  = 1.05457⋅10–34 Js). The magnitude of the angular momentum — which is obtained in every measurement — in these units becomes  I ( I + 1) . If I = 0, then the nucleus has no spin angular momentum and is not active in NMR. The z-component of the spin angular momentum vector is another important operator. This operator is related to the second of two angular momentum quantum numbers. Besides I, there is also m, which specifies the z-component of the spin angular momentum vector: Iˆzψ = m ψ

(1.2)

The quantum number m can attain values ranging between –I and I, in steps of one. We can label the eigenfunctions ψ to the operators Iˆ 2 and Îz with the corresponding quantum numbers, ψI,m. The eigenfunctions are normalized, in the sense that the integral over all space of the product of the eigenfunction ψI,m with its complex conjugate, ψ I*,m , is equal to unity, ∫ ψ I*,mψ I ,mdσ = 1. σ is a variable of integration in the spin space (the spin coordinate). Functions used in quantum mechanics are often complex and the symbol ψ * denotes the complex conjugate of the function ψ, i.e., a corresponding function in which all the imaginary symbols i are replaced by –i (we note that i = −1 ). The normalization requirement is a property of quantum mechanical wave functions. The square of the absolute value of a wave function, ψ *ψ, is related to the probability density of finding the system at a certain value of a relevant coordinate. Consequently, the probability density must integrate to unity over all space. If we integrate, on the other hand, a product of two eigenfunctions corresponding to different values of the quantum number m, the integral vanishes; ∫ ψ I*,mψ I ,m 'dσ = 0 if m ≠ m′, and we say that the functions are orthogonal. The set of functions fulfilling the requirements of normalization and orthogonality is called orthonormal. An alternative way of describing these requirements is through the relation ∫ ψ I*,mψ I ,m′ dσ = δ mm′ , where the symbol δmm′ is called the Kronecker delta and is equal to unity if m = m′ and to zero otherwise. It is often convenient to use the bra-ket (bracket) notation, where the eigenfunctions are treated as unit vectors denoted by a “ket,” |I,m〉. In the bra-ket notation, the normalization condition becomes 〈I,m|I,m〉 = 1, in which the “bra,” 〈I,m|, is associated with ψ *I,m. The star on the first symbol in 〈I,m|I,m〉 is excluded because the bra symbol, 〈|, implies complex conjugation. The integration is replaced by the scalar product of a bra, 〈I,m|, and a ket, |I,m〉.

4

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

For simplicity, we concentrate at this stage on nuclei with the nuclear spin quantum number I = 1/2 (which, in fact, we shall work with throughout most of this book). In this case, there are two possible eigenvalues m of the z-component of the spin: −1/2 and +1/2. The spin eigenfunctions corresponding to these eigenvalues are denoted β and α, respectively. Using the bra-ket formalism, we will use the notation |1/2,–1/2〉 = |β 〉 and |1/2,1/2〉 = |α〉. The orthogonality of the eigenfunctions can be formulated very compactly in the bracket notation: 〈α|β 〉 = 0. If we represent functions as vectors, an operator transforms one vector into another. A set of orthogonal vectors defines a vector space, denoted as Hilbert space, and an operator performs a transformation of one vector in this space into another. Such an operation can be represented by a matrix in the Hilbert space. The elements of such a matrix, called matrix elements of the operator Qˆ , are defined as: Qij = i Qˆ j

(1.3)

The symbols i and j can refer to different eigenstates of the quantum system or to other vectors in the Hilbert space. Qˆ|j〉 is the ket obtained as a result of Qˆ operating on |j〉 and Qij can be considered as a scalar product of the bra 〈i|and that ket. For example, the matrix representation of the z-component of the spin angular momentum in the space defined by the eigenvectors |α〉 and |β 〉 is: 1 0  Iˆz =  2 1 0 − 2

(1.4a)

The matrix is diagonal as a result of our choice of the eigenvectors of the operator Î z as the basis set. We can also obtain matrix representations for the x- and y-components of spin in the basis of |α〉 and |β 〉: 0 1 Iˆx =  1 2   2 0

(1.4b)

0 − i  2 Iˆy =  i  2 0 

(1.4c)

The quantum mechanical operators corresponding to physically measurable quantities are hermitian, i.e., their matrix elements conform to the relation 〈i|Qˆ| j 〉 = 〈 j|Qˆ|i 〉 * (note how this functions for the Iˆy!). The matrix representations of I x , Iˆy, and Iˆz in Equations (1.4) are called the Pauli spin matrices.

Equilibrium and Nonequilibrium States in NMR

5

In quantum mechanics, one often needs to operate on a function (vector) with two operators after each other. In this case, we can formally speak about a product of two operators. The order in which the operators act may be important: we say that the multiplication of operators is in general noncommutative. From this, we can introduce the concept of a commutator of two operators, Qˆ and Pˆ : ˆ ˆ − PQ ˆˆ Qˆ , Pˆ  = QP

(1.5)

For certain pairs of operators, the commutator vanishes and we say that the operators Qˆ and Pˆ commute. The components of spin, Iˆx , Iˆy and Iˆz , do not commute with each other, but each of them commutes with Iˆ 2 . Important theorems about commuting operators state that they have a common set of eigenfunctions and that the corresponding observables can simultaneously have precisely defined values. The proofs of these theorems can be found in the book by Atkins and Friedman (further reading). The spin angular momentum is simply related to the nuclear magnetic moment. In terms of vector components, we can write, for example:

µz = γI Iz

(1.6a)

µˆ z = γ I Iˆz

(1.6b)

where µ z and Iz denote measurable quantities and the quantities with “hats” represent the corresponding quantum mechanical operators. The quantity γI is called the magnetogyric ratio (the notation gyromagnetic ratio is also used). Magnetogyric ratios for some common nuclear species are summarized in Table 1.1, together with the corresponding quantum numbers and natural abundances. TABLE 1.1 Properties of Some Common Nuclear Spin Species

Nucleus 1

H H 13C 14N 15N 17O 19F 31P 23Na 35Cl 79Br 2

Spin quantum number

Natural abundance, %

Magnetogyric ratio, γI/ (107 rad T–1s–1)

1/2 1 1/2 1 1/2 5/2 1/2 1/2 3/2 3/2 3/2

99.98 0.02 1.11 99.63 0.37 0.04 100 100 100 75.53 50.54

26.752 4.107 6.728 1.934 –2.713 –3.628 25.167 10.839 7.080 2.624 6.726

Larmor frequency at 9.4 T, |ωI|/ (108 rads–1) 25.13 3.858 6.321 1.817 2.548 3.408 23.64 10.18 6.652 2.465 6.318

Quadrupole moment, Q/(10–31 m2) — 2.9 — 15.6 — –25.8 — — 103 –82.5 293

6

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

When a nuclear spin is placed in a magnetic field, B, with the magnitude B0, the field interacts with the magnetic moment (the Zeeman interaction). The symbols B and B0 are, strictly speaking, the magnetic induction vector or its magnitude, but we are going to refer to these terms as magnetic field, in agreement with a common practice. The magnetic field in NMR is always assumed to define the laboratory-frame z-axis. In classical physics, the energy of the interaction is written as minus the scalar product of the magnetic moment vector and the magnetic field vector, E = – ⋅ B = –µzB0. In quantum mechanics, the Zeeman interaction is described by the Zeeman Hamiltonian: ˆ = −γ B Iˆ H z I 0 z

(1.7)

The Hamiltonian — or the total energy operator — is a very important operator in quantum mechanics. The eigenfunctions (or eigenvectors) of a quantum system with a general Hamiltonian, Hˆ , fulfill the time-independent Schrödinger ˆ ψ = E ψ or H ˆ|j 〉 = E |j 〉, in the bra-ket notation. For I = 1/2, the equation, H j j j j Zeeman Hamiltonian has two eigenvalues, E1 2 = − 21 γ I B0 and E−1 2 = 21 γ I B0 , corresponding to the eigenvectors |α〉 and| β〉, respectively. The Hamiltonian and its eigenvalues are expressed in angular frequency units, a convention followed throughout this book. The eigenvalues can easily be converted into “real” energy units by multiplying with  . If the magnetogyric ratio is positive (which is the case, for example, for 1H, and 13C nuclei), the α state corresponds to the lowest energy. The difference between the two eigenvalues, ω0 = γIB0, is called the Larmor frequency. (We will use the symbol Ej for energy eigenvalues in angular frequency units and the symbol ω with appropriate index or indices for the energy differences in the same units.) Quantum mechanics does not require the system to be in a specific eigenstate of the Hamiltonian. A spin-1/2 system can also exist in a superposition state, i.e., a superposition of the two states |α〉 and |β 〉. The superposition state is described by a wave function:

ψ = cα α + cβ β

(1.8)

which is subject to the normalization condition, |cα |2 + |cβ | 2 = 1. The coefficients cα and cβ are complex numbers and the squares of the absolute values of the coefficients provide the weights (probabilities) of the two eigenstates in the superposition state. In NMR, the time evolution of quantum systems is of primary interest and the time-dependent form of the Schrödinger equation is very important. Another postulate of quantum mechanics states that the time evolution of the wave function for a quantum system is given by the time-dependent Schrödinger equation:

∂ψ (t) ˆ ψ (t) = −iH ∂t

(1.9)

Equilibrium and Nonequilibrium States in NMR

7

which explains the central role of the Hamilton operator or the Hamiltonian. Note that, following the convention that the Hamiltonian is in angular frequency units, the symbol  that is usually present on the left-hand side of the time-dependent Schrödinger equation is dropped. If the system is initially in one of the eigenstates and the Hamiltonian is independent of time, as is the case for the Zeeman Hamiltonian of Equation (1.7), then the solutions to Equation (1.9) are simply related to the eigenfunctions and eigenvalues of the time-independent Schrödinger equation:

ψ (t) = ψ j exp(−iEjt)

(1.10)

Thus, the system described originally by an eigenstate remains in that eigenstate. The eigenstates are therefore also denoted as stationary solutions to the time-dependent Schrödinger equation or stationary states. The complex exponential factor is called the phase factor (a complex exponential can be expressed in terms of cosine and sine functions, exp(iα) = cosα + i sinα). A more general time-dependent solution can be expressed as a linear combination of functions given in Equation (1.10): I

ψ (t) =

∑c ψ m

exp(−iEmt)

(1.11a)

exp(−iEmt) I , m

(1.11b)

I ,m

m=− I

or, using the bracket notation, I

ψ (t) =

∑c

m

m=− I

Another important quantity required for the discussion of quantum systems is the expectation value. Through still another of the postulates of quantum mechanics, the expectation value of an operator corresponding to an observable is equal to the average value of a large number of measurements of the observable under consideration for a system not in the eigenstate of that operator. Consider, for example, the x-component of the nuclear magnetic moment; the expectation value of this quantity at time t is defined by:

∫

µˆ x (t) = ψ *(t)µˆ xψ (t)dσ

(1.12a)

µˆ x (t) = ψ (t) µˆ x ψ (t)

(1.12b)

or

8

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

We note that the expectation value of the x-component of the magnetic moment, 〈 µˆ x 〉(t), is explicitly time dependent, while the operator µˆ x is not. Thus, the time dependence of the expectation value originates from the time dependence of the wave functions. This way of expressing the time dependence is denoted the Schrödinger representation. Using the definitions of Equation (1.11a) and Equation (1.11b), as well as the relation µˆ x = γ I Iˆx , we can evaluate Equation (1.12b): I

µˆ x (t) = γ I

I

∑ ∑c

* m′ m

m

m′

c

(

I , m′ Iˆx I , m exp i(Em′ − Em )t

)

(1.13)

From elementary properties of angular momentum (see, for example, Atkins and Friedman (further reading)), we know that expressions for timeindependent matrix elements 〈 I , m′|Iˆx|I , m〉 = ∫ ψ I*,m′ Iˆxψ I ,mdσ vanish, unless m′ = m ± 1. For the case of I = 1/2, we thus have nonvanishing elements for m′ = 1/2 if m = −1/2 or vice versa, corresponding to the matrix representation of Î x in Equation (1.4b). When this condition is fulfilled, we can see that the argument of the exponential function becomes ±iω0t, where ω0 = –γIB0 is the Larmor frequency. Noting both these relations, one can demonstrate (see Slichter’s book (further reading) or do it as an exercise) that the expectation value of µˆ x oscillates with time at the Larmor frequency. In the same way, one can show that the expectation value of µˆ = γ Iˆ y

I y

also oscillates at the Larmor frequency, and the expectation value of the z-component is constant with time. A detailed analysis (see Slichter’s book) shows that the quantum mechanical expectation value of the nuclear magnetic moment vector operator ˆ behaves in analogy with a vector of length γI/2 moving on a conical surface around the z-axis — the direction of the magnetic field — in analogy to the classical Larmor precession. Another important result is that the angle between the conical surface and the magnetic field can have an arbitrary value and that the timeindependent z-component of the magnetic moment of a nuclear spin in a superposition state can hold any arbitrary value between –γI/2 and + γI/2. We emphasize that this result is consistent with the notion that the spins can exist in superposition states and are not confined to pointing parallel or antiparallel to the external field. In the same vein, we cannot associate an individual spin in a superposition state with any of the two energy eigenstates (energy levels) of the Hamiltonian, even though the motion is clearly related to the energy difference between the Zeeman states.

1.2

Ensembles of Spins: the Density Operator

So far, we have only considered the properties of one single spin. The concepts of relaxation and equilibrium are closely connected to the behavior of macroscopic samples of spins. A theoretical tool we need to use is that of an

Equilibrium and Nonequilibrium States in NMR

9

ensemble of spins — a large collection of identical and independent systems. For simplicity, we deal here with an ensemble of spin-1/2 particles, interacting with the magnetic field through the Zeeman interaction but not interacting with each other. The density operator method is an elegant way to deal with a very large number (on the order of 1021) of quantum systems, corresponding to a macroscopic sample. We present here a very brief summary of the technique and recommend the books in the further reading section (especially the books by Abragam, Ernst et al., Goldman, Levitt, and Slichter) for more comprehensive presentations. The density operator approach allows us to calculate expectation values of operators for ensembles of quantum systems rather than for individual systems. Let us assume that a certain individual spin is described by a superposition state wave function, according to Equation (1.8). We disregard for the moment the time dependence. The expectation value of an operator Qˆ is for that spin given by: Qˆ =

I

I

∑ ∑c

* m′ m

m=− I

c

I , m′ Qˆ I , m (1.14)

m′=− I

= cα* cα α Qˆ α + cβ* cα β Qˆ α + cα* cβ α Qˆ β + cβ* cβ β Qˆ β in analogy with the case of the x-component of the magnetic moment, Equation (1.12b). If we wish to make a similar calculation for another spin, the coefficients cα and cβ will be different but the matrix elements of the operator Qˆ will be the same. Averaging over all spins in the ensemble, we obtain: Qˆ = cα* cα α Qˆ α + cβ* cα β Qˆ α + cα* cβ α Qˆ β + cβ* cβ β Qˆ β =

∑ ∑c

* m′ m

m

=

c

m′ Qˆ m =

m′

∑ ∑ m ρˆ m′ m

∑ ∑ρ m

mm′

m′ Qˆ m

(1.15)

m′

m′ Qˆ m = Tr( ρˆ Qˆ )

m′

The bar over the products of coefficients denotes ensemble average. Here, we have introduced the symbol ρˆ , which denotes the density operator, with the matrix representation 〈m|ρˆ |m′〉 = ρm ,m′ = cm* ′ cm . Tr( ρˆ Qˆ ) represents taking the trace (summing the diagonal elements in a matrix representation) of the product of the two operators or the two matrices. The preceding definition shows that the density operator is hermitian. If the density operator for an ensemble is known, then the expectation value of any operator corresponding to observable quantities can be computed. We may wish to calculate the expectation value as a function of time. This can be done by expressing the superposition states in terms of time-dependent

10

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

coefficients (and absorbing the phase factors exp(–iEt) into them). More practically, we can calculate the time dependence of the density operator directly, using the Liouville–von Neumann equation: d ˆ ρ(t) = −i  Hˆ (t), ρˆ (t)  = i  ρˆ (t), Hˆ (t)  dt

(1.16)

where the concept of a commutator of the two operators, introduced in Equation (1.5), is used. The Liouville–von Neumann equation can be derived in a straightforward way from the time-dependent Schrödinger equation. Using time-dependent density operator ρˆ (t) , the time-dependent form of the expectation value of Equation (1.15) can be written as: Qˆ (t) =

∑ ∑c

* m′

m

(t)cm (t) m′ Qˆ m =

m′

∑ ∑ρ m′

m

mm′

(t) m′ Qˆ m

(1.17)

= Tr( ρˆ (t)Qˆ ) The matrix representation of the density operator is called the density matrix. If we assume that the time dependence resides in the density matrix rather than in the operator, the formulation of the time-dependent expectation values in Equation (1.17) is denoted as the Schrödinger representation, in analogy with the single-spin case of Equations (1.12). The elements of the density matrix have a straightforward physical interpretation. The diagonal elements, ρmm, represent the probabilities that a spin is in the eigenstate specified by the quantum number m, or the relative population of that state. At the thermal equilibrium, these populations are given by the Boltzmann distribution:

ρmm =

exp(− Em/kBT )

∑ exp(−E /k T) j

(1.18)

B

j

T is the absolute temperature and kB is the Boltzmann constant, kB = 1.38066⋅10–23 JK–1. Nuclear spins are quantum objects and their distribution among various quantum states should in principle be obtained using the Fermi–Dirac statistics (for half-integer spins) or by Bose–Einstein statistics (for integer spins) rather than from Equation (1.18). However, for anything but extremely low temperatures, the Boltzmann statistics is an excellent approximation. The energy differences involved in NMR are tiny, which results in very small population differences. For an ensemble of N spin-1/2 particles with a positive magnetogyric ratio, we can write: eq nαeq/N = ραα =

 1 1   1  1  exp  1 γ I B0/kBT   ≈  1 + 1 (γ I B0/kBT ) =  1 + bI    2  2  2 2 2  2 (1.19a)

Equilibrium and Nonequilibrium States in NMR

11

and eq nβeq/N = ρββ =

 1  1 1 1  1  1  exp  − γ I B0/kBT   ≈  1 − (γ I B0/kBT ) =  1 − bI    2   2 2 2  2  2 (1.19b)

where we expand the exponential in a power series, retain only the linear term and introduce the quantity bI = γ I B0/kBT , called the Boltzmann factor. N is the total number of spins. Retaining only the linear term is valid as long as γ I B0 << kBT , a condition easily fulfilled for anything but extremely low temperatures. The approximation of retaining only the linear term is called the high temperature approximation. For protons, which have the magnetogyric ratio γH = 26.7522⋅107 T–1s–1, at 300K and in the magnetic field of 9.4 T (corresponding to the 1H frequency of 400 MHz), we obtain nαeq/N = 0.500016 and nβeq/N = 0.499984. At 4K, the liquid helium temperature, the corresponding numbers are nαeq/ N = 0.5024 and nβeq/N = 0.4976. Clearly, the natural Boltzmann polarization of the two nuclear spin levels is very low — the origin of the poor sensitivity of NMR — compared to other spectroscopic techniques. The population difference between the spin energy levels determines the expectation value of the z-component of the ensembleaveraged nuclear magnetic moment, which we call the magnetization vector, of an ensemble of spins. This can be seen (the reader is recommended to prove it as an exercise) by inspecting Equation (1.15) and recognizing the fact that the matrix representation of µˆ z or Iˆz (see Equation 1.4a) has only diagonal elements. The off-diagonal elements of the density matrix are called coherences. The coefficients cm can be written as a product of an amplitude |cm| and a phase factor exp(iam). The coherence between the eigenstates m and m′ is given by: cm* ′ cm = cm* cm′ exp(i( am − am′ ))

(1.20)

The coherences are closely related to the magnetization components perpendicular to the magnetic field (the transverse magnetization). The phase factor exp(i(am – am′)) for an individual spin specifies the direction of the transverse magnetic moment. For a large number of spins at thermal equilibrium, there is no physical reason to assume any of the directions perpendicular to the field to be more probable than any other, which amounts to a random distribution of the phase factors and vanishing coherences.

1.3

Simple NMR: the Magnetization Vector

According to the previous section, an ensemble of noninteracting nuclear spins at the thermal equilibrium can be represented by a magnetization vector, M, oriented along the direction of the external magnetic field. As we

12

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

shall see later, the spins do interact with each other, but the interactions are usually very weak compared to the Zeeman energies. In addition, the interactions tend to average to zero because of molecular motions in isotropic fluids. For all practical purposes, the spins in isotropic liquids can be considered as noninteracting if their NMR spectra do not show spin–spin splittings ( J couplings). The magnitude of the magnetization vector at thermal equilibrium, M0, is proportional to the Boltzmann factor and depends, in addition, on the magnetic moment of an individual spin and on the number of spins: M0 =

Nγ I2  2 I ( I + 1)B0 3 k BT

(1.21)

The magnetization vector is a macroscopic quantity and its motion can be described using classical physics. Classically, if a magnetic moment is not aligned along the magnetic field, it will precess around the field direction, the same motion that we found earlier quantum mechanically for the magnetic moment of an individual spin and which we refer to as Larmor precession. The concept of a magnetization vector is very useful for describing NMR experiments in systems of noninteracting spins. We can use it, for example, to describe the effect of radiofrequency pulses. To describe an NMR experiment, we need to consider the presence of a static magnetic field in the zdirection with the magnitude B0, as well as the time-dependent magnetic field corresponding to the magnetic component of electromagnetic radiation (the radiofrequency field), B1. The role of the radiofrequency field is simplest to consider in a frame rotating around the B0 field with the angular velocity corresponding to the radiofrequency. The reader can find the derivation and discussion of the rotating frame in several books, e.g., Slichter or Levitt (further reading). For our purposes, it is sufficient to say that if a radiofrequency field is applied near resonance (meaning that the radiofrequency is close to the Larmor frequency), the nuclear spins move as if it were the only field present — i.e., they precess around it with the angular velocity γIB1. In this rotating frame, the applied radiofrequency field appears to be static. If we keep the radiofrequency field switched on for a time τ, the magnetization will rotate by the angle γIB1τ. Adjusting the magnitude of the radiofrequency field, and/or the time during which we apply it in such a way that the angle becomes π/2, we obtain a 90° pulse or a π/2 pulse, which has the effect of turning the magnetization, originally in the z-direction to the transverse plane, according to Figure 1.1. Setting the condition γIB1τ = π results in a 180° pulse or π pulse. Based on classical equations of motion for the dynamics of the magnetization vector, the transverse magnetization in the rotating frame will, after a 90° pulse on resonance, retain its direction and magnitude. If the radiofrequency pulse is not exactly on resonance, i.e., if there is a frequency offset, ωoff, between the applied radiofrequency and the Larmor frequency,

Equilibrium and Nonequilibrium States in NMR

13

z

M0

y B1

x FIGURE 1.1 Illustration of the effect of radiofrequency pulses, here a 90° pulse.

the transverse magnetization after a 90° pulse is expected to precess in the rotating frame with the frequency corresponding to the offset. Thus, the rotating frame serves to make the frequency offset the only precession frequency. Due to chemical shifts, the offset will vary for chemically inequivalent nuclear spins of the same species, e.g., for inequivalent protons. The motion of the magnetization vector generates an oscillating signal in the detector of the NMR spectrometer, the free induction decay or FID. The FID in this simple picture is expected to last forever, which we know is not in agreement with experimental facts. The origin of the decay of the FID is nuclear spin relaxation. The description of even the very simplest NMR experiment, the detection of an FID after a 90° pulse, requires that we take into consideration two types of motion: the coherent motion or precession around the effective fields and the incoherent motion or relaxation. A very simple model containing these two elements is described by the Bloch equations, called so after one of the inventors of NMR, who proposed them in a seminal paper from 1946.1 The Bloch equations can be written in the following, slightly simplified, form: dM z M0 − M z = dt T1

(1.22a)

dMx M = M yω off − x dt T2

(1.22b)

dM y dt

= − Mxω off −

My T2

(1.22c)

Bloch equations are phenomenological, i.e., they aim at a simple description of the observed NMR phenomenon, without requirements of a strict

14

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

derivation. We will show later that the relaxation behavior in the form of the Bloch equations can be derived in some situations and more complicated relaxation phenomena are predicted in other cases. Equation (1.22a) describes the time variation of the longitudinal (along the external field) component of the magnetization vector. The equation predicts the magnetization component along B0 to relax exponentially to its equilibrium value, M0. The time constant for that process is called the spin-lattice or longitudinal relaxation time and is denoted T1. The rate of exponential recovery of Mz to equilibrium is given by the inverse of T1, sometimes denoted R1, and called spin-lattice relaxation rate (the notation rate constant rather than rate would be more logical but is rarely used). We will use T1 and R1 to describe longitudinal relaxation throughout this book. The solution of the Bloch equation for Mz, after the initial inversion of the magnetization (corresponding to the application of a 180° pulse at the time t = 0) can be written as: Mz (t) = M0 (1 − 2 exp(−t/T1 ))

(1.23)

The process of recovery of Mz is illustrated in Figure 1.2. The reader is encouraged to prove that Equation (1.23) is a solution to Equation (1.22a). Equation (1.22b) and Equation (1.22c) describe the motion of the transverse components of the magnetization vector. The first part of the expressions corresponds to the coherent motion of M in the rotating frame. The second part introduces the concept of the transverse, or spin–spin relaxation time, T2, describing the exponential decay of the xy-magnetization to its equilibrium value of zero. One interesting observation we can make in Equation (1.22b) and Equation (1.22c) is related to the units. Clearly, the factors multiplied by the magnetization components in the two terms should have the same dimensions. Because the natural unit for the angular frequency is radians per second, the relaxation rate, or the inverse relaxation time, R2 = 1/T2, should indeed also

FIGURE 1.2 Illustration of the recovery of the Mz magnetization after the 180° pulse in the inversion-recovery experiment.

15

Mx,y (arbitrary unit)

Equilibrium and Nonequilibrium States in NMR

Time FIGURE 1.3 The decay of on-resonance transverse magnetization as a function of time.

be expressed in these units. Usually, relaxation times are given in seconds (the rates are given in s–1), which tacitly implies that radians can be omitted; we note in parenthesis that the radian is considered a dimensionless unit in physics. Assuming that the pulse is on-resonance, the evolution of the transverse components of the magnetization vector after a 90° pulse is expressed as: Mx , y = M0 exp(−t/T2 )

(1.24)

The decay of the transverse magnetization for the on-resonance situation is illustrated in Figure 1.3. We recall that the NMR spectrum for a system of noninteracting spins is the Fourier transform of the transverse magnetization (see for example, Ernst et al. or Levitt, further reading). The Fourier transform of an exponential decay is a Lorentzian centered at zero frequency, with the full width at half-height (in Hertz) equal to ∆v = 1/πT2 (compare Figure 1.4). The reason for introducing two different relaxation times is that the return to the equilibrium is a physically different process for the longitudinal and transverse magnetization components. The longitudinal relaxation changes the energy of the spin system and thus involves the energy exchange between the spins and the other degrees of freedom in the surrounding matter (the notion of spin-lattice relaxation originates from early NMR work on solids, in which

(πT2)−1

−50

−25

0 Frequency(Hz)

25

50

FIGURE 1.4 Relationship between the Lorentzian lineshape and the transverse relaxation time constant, T2.

16

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

these other degrees of freedom were those of the crystal lattice). The transverse relaxation involves, on the other hand, the loss of phase coherence in the motion of individual spins.

1.4

Coupled (Interacting) Spins: the Product Operator Formalism

The physical description of a spin system using magnetization vectors is useful for an ensemble of noninteracting spins. For the case of coupled spins, more refined tools are necessary to describe NMR experiments and NMR relaxation processes. These tools have their basis in the concept of the density operator. In Section 1.2, we introduced the density operator by means of its matrix representation, the density matrix. For ensembles of interacting spins, working with the matrix representation of the density operator quickly becomes quite difficult to handle; therefore, it is often useful to work directly with the expansion of the density operator in another type of vector space, called the Liouville space. The concept of the Liouville space is discussed in the book by Ernst et al. (further reading), among others, and in a review by Jeener.2 A basis set in the Liouville space can be formulated in terms of the basis set in the Hilbert space in the following way. Consider the case of our isolated spin 1/2 nucleus, with its basis set of |α〉 and |β〉. As we discussed earlier, the scalar products of a bra and a ket, such as 〈α | β〉, are numbers. We can also define outer products of these vectors, in which we have a ket to the left and a bra to the right. Four such constructs are possible:

α α , α β , β α , β β

(1.25)

These objects are operators, which can be illustrated by the following example:

α β β = α 1= α

(1.26)

The operator |α〉 〈β | acts on the ket |β 〉. 〈β | β 〉 is a number (unity) and we thus obtain that the result of the operation is another ket, |α〉. The operators created as outer products of the basis vectors in the Hilbert space can be used to form an operator basis set in the Liouville space. If the dimensionality of the Hilbert space is n (n = 2 in our example of an isolated spin 1/2, with |α〉 and |β 〉 as the basis vectors), then the dimensionality of the corresponding Liouville space is n2 (which in our case is four, corresponding to the four basis vectors |α〉 〈α |, | α〉 〈β |, | β 〉 〈α |, |β 〉 〈β |). The operators expressed as outer products will be an exception from the rule of decorating the operators with “hats.” It is easy to obtain the matrix representations of the four preceding operators in the Hilbert space. For example, the matrix representation of the |α〉   〈α operator is  10 00 . The reader is advised to confirm this and to derive the matrices corresponding to the other three operators. Comparing the matrix

Equilibrium and Nonequilibrium States in NMR

17

representations of the operators of Equation (1.25) with the Pauli spin matrices introduced in Equations (1.4), we can note that: 1 2

(α

)

α + β β = 21 1ˆ op

1 2

(α

α − β β = Iˆz

1 2

(α

β + β α = Iˆx

i 2

(β

α − α β = Iˆy

(1.27a)

)

(1.27b)

)

(1.27c)

)

(1.27d)

where we have introduced the unit operator, 1ˆ op, with the property that it leaves whatever function or vector coming after it unchanged. The unit operator is represented by a unit matrix, a matrix with ones on the diagonal and zeroes elsewhere. Equation (1.27a) through Equation (1.27d) imply an orthogonal transformation of one set of vectors in the Liouville space into another. Once we have defined an appropriate Liouville space basis, we can expand any other operator in that space. For example, the density operator at thermal equilibrium for an ensemble of isolated spins can be expressed as: 1 1 ρˆ eq = 1ˆ op + bI Iˆz 2 2

(1.28)

where the Boltzmann factor, bI, was defined together with Equation (1.19b). The idea of expanding the density operator into an operator basis set related to the spin operators is easily generalized to more complicated spin systems. For a system of two spins, I and S, with the components, Iˆx , Iˆy , Iˆz and Sˆ , Sˆ , Sˆ , we can form an appropriate basis set by including the unit x

y

z

operator for each spin. The product operator basis for a two-spin system will thus consist of 4 × 4 operators, given by the product of one operator for the spin I and one operator for spin S, with suitable normalization. This product operator basis will consist of 21 1ˆop , Iˆx , Iˆy , Iˆz , Sˆ x , Sˆ y , Sˆ z , 2 IˆxSˆ x , 2 IˆxSˆ y , 2 IˆxSˆ z , 2 Iˆy Sˆ x , 2 Iˆy Sˆ y , 2 Iˆy Sˆ z , 2 Iˆz Sˆ x , 2 Iˆz Sˆ y , 2 Iˆz Sˆ z , a total of 16 operators. A two-spin system is characterized by the occurrence of four eigenstates of the Zeeman Hamiltonian, |αα〉, |αβ 〉, |βα〉, |ββ 〉, implying the dimensionality of four for the Hilbert space and the dimensionality of 42 = 16 for the Liouville space. Clearly, we retain the dimensionality of the Liouville space moving between the basis set consisting of ket-bra products |αα〉 〈αα |, etc. and the product operator basis. Contrary to the density matrix calculations, the product operators provide a method to easily describe NMR experiments and are invaluable for evaluating NMR pulse sequences on interacting spins. The

18

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

product operator formalism has in particular been used for improving and designing new experiments. We will explore this further in the chapters describing experimental techniques and applications. In the same way in which one can define operators in the Hilbert space, it is possible to construct their analogues in the Liouville space. These are called superoperators. The superoperator analogue of the Hamiltonian is called Liouville superoperator or Liouvillian, Lˆˆ (we will use the “double hat” symbol for superoperators). It is defined as commutator with the Hamiltoˆ , ] . Operating with the Liouvillian on another operator amounts nian, Lˆˆ = [ H thus to taking the commutator of the Hamiltonian with that operator. This superoperator formalism can be used to rewrite the Liouville–von Neumann equation (Equation 1.16) in the form: ˆ d ˆ ρ = −i[ Hˆ , ρˆ ] = −iLˆ ρˆ dt

(1.29)

Equation (1.29) illustrates one of the reasons to use the superoperator formalism: it allows very compact notation. Another important superoperator is the relaxation superoperator, which operates on the density operator and describes its evolution towards equilibrium. In the same way in which operators are represented in the Hilbert space by matrices, the superoperators have matrix representations in the Liouville space denoted as supermatrices. The concept of relaxation supermatrix is very important in relaxation theory and we will come back to it in Chapter 4. Yet another type of superoperators are various rotation superoperators introduced in Chapter 7. In some situations, we need to consider more complicated evolution of the density operator and the Liouville space formalism is then very useful; we will see examples of that application of the formalism in Chapter 10 and Chapter 15.

References 1. Bloch, F. Nuclear induction, Phys. Rev., 70, 460–474, 1946. 2. Jeener, J. Superoperators in magnetic resonance, Adv. Magn. Reson., 10, 1–51, 1982.

2 Simple Relaxation Theory

CONTENTS 2.1 An Introductory Example: Spin-Lattice Relaxation...............................19 2.2 Elements of Statistics and Theory of Random Processes .....................22 2.2.1 Stochastic Variables.........................................................................22 2.2.2 Stochastic Functions of Time.........................................................25 2.3 Time-Dependent Perturbation Theory and Transition Probabilities in NMR Relaxation Theory.................................................29 2.4 Predictions of the Simple Model...............................................................32 2.5 Finite-Temperature Corrections to the Simple Model...........................37 References ............................................................................................... 38

In order to establish a foundation for relaxation theory, we will in this chapter provide a background by introducing the concepts of spin-lattice and spin–spin relaxation. We begin by looking at a simple example, which captures the basic principles of the nuclear spin relaxation without being directly applicable to any real physical situation. We use this simple example to introduce the important basic concepts within the theory of random processes and then to calculate transition probabilities driven by random processes. We look briefly at the prediction of the simple model and introduce the important finite temperature corrections to the theory.

2.1

An Introductory Example: Spin-Lattice Relaxation

Among the two types of relaxation processes, the spin-lattice relaxation is easier to explain in simple conceptual terms, and therefore we will start with describing this process. Let us consider a system of I = 1/2 spins with a positive magnetogyric ratio in a magnetic field B0. We deal thus with a two-level system with populations nα and nβ , with the energy spacing ∆E = – ω0 = γ B0 and with the Larmor frequency ω 0. At thermal equilibrium, the relative populations of the two levels are given by Equation (1.19a) and Equation (1.19b).

19

20

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

Let us assume that the spin system is not in equilibrium. The nonequilibrium situation can be produced in different ways. One way is to change the magnetic field quickly. The equilibrium population distribution at the original field does not correspond to the equilibrium at the new field. An everyday example of such an experiment is putting an NMR sample into the NMR magnet. A better controlled variety of such an experiment is the field-cycling experiment, in which the magnet current is switched rapidly and to which we will return in Chapter 8. An important feature of an experiment of this kind is the fact that the Hamiltonian — and thus the Zeeman splitting — changes immediately, while the density operator — for example, populations — requires some time to adjust. This is the simplest example of spin-lattice relaxation. Another way of creating nonequilibrium states in NMR, mentioned in Section 1.3, is to use radiofrequency pulses. For example, a 180° pulse (or a π -pulse) inverts the populations in a two-level system. We assume that the changes of the populations of the two levels follow the simple kinetic scheme: dnα = (nβ − nβeq )Wβα − (nα − nαeq )Wαβ = WI (nβ − nβeq − nα + nαeq ) dt

(2.1a)

dnβ = WI (nα − nαeq − nβ + nβeq ) dt

(2.1b)

Equation (2.1a) and Equation (2.1b) imply that the α - and β - levels are populated (and depopulated) by first-order kinetic processes, with the rates (in the sense of chemical kinetics) proportional to the deviations of the populations from the equilibrium values. The proportionality constants are transition probabilities, assumed for the time being equal in both directions, Wβα = Wαβ = WI. We will return to this point in Section 2.5. Instead of discussing the changes of the populations nα and nβ , we can introduce new variables, the difference in populations, n = nα − nβ , and the sum of the populations, Ν = nα + nβ . In terms of these variables, Equation (2.1a) and Equation (2.1b) can be rewritten as: dN =0 dt

(2.2a)

dn = −2WI (n − neq ) dt

(2.2b)

Equation (2.2a) tells us that the total number of spins is constant; Equation (2.2b) says that the population difference returns to equilibrium in an exponential process. We note that the population difference is proportional to the longitudinal component of the magnetization vector, n(t) ~ γ I 〈 Iˆz 〉(t) = Mz (t) . By inspection of the first of the Bloch equations, Equation (1.21a), we can relate the T1 to WI according to: T1–1 = 2WI

(2.3)

Simple Relaxation Theory

21

The relaxation rate is thus proportional to a transition probability. The transitions giving rise to NMR relaxation are nonradiative, i.e., they do not arise through emission or absorption of radiation from radiofrequency fields. Instead, they occur as a result of weak magnetic interactions, with the origin in the sample, if those oscillate in time with frequency components at the Larmor frequency. The weakness of the relevant interactions for spin 1/2 nuclei results in small transition probabilities and the spin-lattice relaxation processes being slow, typically on the millisecond to second time scale. More specifically, the time dependence of the interactions has its origin in random molecular motions. The effect of molecular motions can be explained as follows. Many interactions in NMR are anisotropic, i.e., depend on the orientation of the spin-carrying molecule in the magnetic field. A prominent example, to be discussed in detail in the next chapter, is the dipole–dipole interaction. Thus, the interaction changes when the molecule reorients. We concentrate at this stage on the case of ordinary, isotropic liquids. As opposed to the situation in the gas phase, molecules in a liquid are surrounded by neighbors and are not able to rotate freely. Rather, the rotational motion of a moleculefixed axis in a molecule immersed in a liquid can be pictured as a sequence of small angular steps (a random walk on the surface of a sphere). This is illustrated in Figure 2.1. The combination of the random walk and the anisotropic interactions gives rise to Hamiltonians for nuclear spins varying randomly with time. The effect of these random, or stochastic, interactions is to cause transitions, which are intimately connected with the nuclear spin relaxation processes. Therefore,

FIGURE 2.1 Illustration of a random walk on the surface of a sphere.

22

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

we need to expand on this issue and explain how this happens in two steps. Because the random nature of the interactions is essential, we introduce first some basic ideas and concepts from the theory of random processes. Second, we present a quantum mechanical treatment of transition probabilities caused by random motions.

2.2

Elements of Statistics and Theory of Random Processes

The results of many physical experiments on molecular systems are best described by using statistical methods. This is because many processes are random in nature and we need a way to characterize expectation values and averages. In this section, we provide an introduction to the theory of random processes. A more complete description of the subject can be found, for example, in the book of Van Kampen (see the further reading section in the Preface). 2.2.1

Stochastic Variables

Consider a quantity that can be measured and assigned a numerical value x. Assume that the numerical values vary within a certain interval between different realizations of the measurement in an unpredictable way. We then call X a stochastic variable. The values, x, that X adopts are called numerical realizations. An example from daily life is the length of an individual person, in which X is a symbolic notation for the length of a person and x is the value of the length. This, of course, varies stochastically within a certain group of people. An example relevant for NMR is the orientation of a molecule-fixed vector with respect to the laboratory z-axis. The orientation can be specified in terms of an angle θ, as indicated in Figure 2.2, which can be measured (at least in principle) and thus assigned a numerical realization. A stochastic variable can be described by a probability density, p(x): p( x)dx = P( x ≤ X ≤ x + dx)

(2.4)

where P(x ≤ X ≤ x + dx) denotes the probability of the numerical realization taking on a value within the indicated, infinitesimally small interval between x and x + dx. Rather than working in terms of infinitesimal intervals, we can use an integrated form: x2

∫ p(x)dx = P(x ≤ X ≤ x ) 1

2

(2.5)

x1

which describes the distribution of values x that the variable X adopts. The probability density can be established by a long series of measurements. It is easy to imagine this being done for the length of an individual

Simple Relaxation Theory

23

FIGURE 2.2 Illustration of the orientation of a molecule-fixed vector with respect to the laboratory coordinate frame. The orientation can be specified in terms of the angles θ and φ.

in a certain population, but perhaps not so for the second example described previously. For that example — the orientation of a molecule-fixed vector in isotropic liquid with respect to an external frame — one would rather make an assumption that the distribution of the angles is isotropic, i.e., that all angles are equally probable. The value of the probability density is then determined by the normalization condition, i.e., by the requirement that the integral of the probability density over all angles must yield unity. We will return to this point in Chapter 6. Stochastic variables often occur in pairs. For example, we can measure the length and the weight of an individual within a certain population. We call such two variables X and Y and denote the corresponding numerical realizations x and y. We introduce a probability density in two dimensions: p( x ; y)dxdy = P( x ≤ X ≤ x + dx ; y ≤ Y ≤ y + dy)

(2.6)

where the symbol “;” means “and.” Alternatively, the corresponding integrated form can be used: y2 x2

∫ ∫ p(x; y)dxdy = P(x

1

≤ X ≤ x2 ; y1 ≤ Y ≤ y2 )

(2.7)

y1 x1

If the two stochastic variables are statistically independent (uncorrelated), then: p( x ; y) = p( x) p( y)

(2.8)

24

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

An important property of a stochastic variable is its average value (mean) and this is given by: ∞

〈X 〉 =

∫ xp(x)dx

(2.9)

−∞

We prefer to use the symbol 〈X 〉 rather than X for average; the bar is used when the other notation can be confusing. Analogously, we can define an average value, expectation value, of a function of X. For example, the average value of Xn (the n-th power of X), called the n-th moment, is defined as: ∞

mn = 〈X n 〉 =

∫ x p(x)dx n

(2.10)

−∞

Another important average is the average of (X – c)n, where c is a constant. This is called the n-th moment around c. Moments around mean values are often used in statistics. They are denoted µn, and the second moment, µ2, is particularly important:

µ2 = 〈(X − m1 )2 〉 = 〈X 2 〉 − 2 m1 〈X 〉 + m12 = 〈X 2 〉 − m12 = σ 2

(2.11)

and is called variance. The square root of the variance is called standard deviation, σ. Average values can also be defined for cases involving more than one stochastic variable, e.g., for a product: ∞ ∞

m11 = 〈XY 〉 =

∫ ∫ xyp(x; y)dxdy

(2.12)

−∞ −∞

It can be useful to express p(x;y) as a product: p( x ; y) = p( x) p( x|y)

(2.13)

where p(x | y) means the probability density for Y acquiring the value y, provided that X assumes the value x. This is called conditional probability density. If X and Y are statistically independent, then p(x|y) = p(y) and: m11 = 〈XY 〉 = 〈X 〉〈Y 〉

(2.14)

Simple Relaxation Theory

25

We can also define a mixed second moment:

µ11 = 〈(X − 〈X 〉)(Y − 〈Y 〉)〉 = m11 − m10m01

(2.15)

which vanishes if X and Y are statistically independent. A convenient way to express the correlation between two variables is to define a correlation coefficient, ρ, between X and Y. This is defined as:

ρ=

µ11 σ Xσ Y

(2.16)

where σX and σY are the standard deviations for the two stochastic variables. For statistically independent X and Y, ρ = 0. For the opposite limiting case, X = Y, it is easily seen that µ 11 = µ 2 = σ 2, and thus we have ρ = 1. 2.2.2

Stochastic Functions of Time

In NMR relaxation theory, stochastic processes (stochastic functions of time) are important. As was mentioned in the preceding section, the combination of random walk and anisotropic interactions leads to stochastic interactions. Stochastic processes, Y(t), are those that give rise to a time-dependent stochastic variable, which means a quantity that at every point, t, in time behaves as a stochastic variable. The stochastic process is characterized by a probability density, p(y,t), in general also dependent on time. An example of such a process might be the depth of water at a certain point along an ocean beach, measured on a windy day. Because of the waves, the water level changes with time, within certain limits, and in a random way. The NMR-relevant example mentioned before — the angle between a molecule-fixed vector and the external magnetic field — is a typical stochastic function of time because of random motions (compare Figure 2.1). The average value of Y(t) at time t is defined: ∞

〈Y (t)〉 =

∫ yp(y, t)dy

(2.17)

−∞

The properties of a stochastic function, Y(t), at different times t are in general not independent. We will investigate the correlation between Y(t) at times t1 and t2. It is rather easy to imagine that such a correlation exists if the time points t1 and t2 are close to each other on the timescale defined by the random oscillation of the process Y(t). This is shown in Figure 2.3a and Figure 2.3b, illustrating a rapidly decaying and a more persistent correlation. We introduce p(y1,t1;y2,t2), the probability density for Y(t) acquiring value y1

26

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

Time

Time (a)

Time

Time (b) FIGURE 2.3 Illustration of a stochastic function of time and the corresponding correlation function for a rapidly-vanishing correlation, or short correlation time (a), and a more persistent correlation, long correlation time (b).

Simple Relaxation Theory

27

at t1 and y2 at t2. Also in this case, we can use the concept of conditional probability: p( y1 , t1 ; y2 , t2 ) = p( y1 , t1 )p( y1 , t1|y2 , t2 )

(2.18)

A stochastic process is called stationary if the probability density p(y,t) does not vary with time. If this is the case, the conditional probability density simplifies to: p( y1 , t1|y2 , t2 ) = p( y1 , 0|y2 , t2 − t1 ) = p( y1|y2 , τ )

(2.19)

where we have introduced the length τ = t2 − t1 of the time interval between t1 and t2. For the average value of a product of a stationary Y(t) at different times, t1 and t2, we have, from Equation (2.12):

∫ ∫ y (t )y (t )p(y , t )p(y , t |y , t )dy dy = ∫ ∫ y y p(y , 0)p(y , 0|y , t − t )dy dy

〈Y (t1 )Y (t2 )〉 =

1

1

1 2

2

2

1

1

1

1

1

2

1

2

2

1

2

1

1

2

(2.20)

2

The expression for the average value depends only on the time difference τ = t2 − t1 and we can define: 〈Y (t1 )Y (t2 )〉 = G(t2 − t1 ) = G(τ )

(2.21)

The quantity G(τ) is called time-correlation function (tcf). Because it correlates a stochastic process with itself at different points in time, it is called the autocorrelation function. The autocorrelation functions for the random processes in Figure 2.3a and Figure 2.3b are also shown there. Cross-correlation functions can also be defined and we will return to such functions later. In relaxation theory, we often deal with complex functions Y(t). For such cases, the definition of the autocorrelation function must be modified slightly: G(τ ) = 〈Y (t)Y ∗ (t + τ )〉

(2.22)

For G(τ) defined in this way, we have: G(τ ) = G∗ (τ ) = G(−τ )

(2.23)

which means that the time-correlation function is real and an even function of time. For τ = 0, we obtain then: G(0) = 〈Y (t)Y * (t)〉 = 〈|Y (t)|2 〉 = σ 2

(2.24)

That is, the autocorrelation function of Y at zero time is equal to the variance of Y.

28

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

Let us further assume that 〈Y(t)〉 = 0. This assumption does not really cause any loss of generality for a stationary process because we can always subtract the time-independent average from our stochastic function of time. For the limit of very long time, τ → ∞, it is reasonable to assume that Y(t) and Y(t + τ) become uncorrelated and we can write: lim G(τ ) = Y (t)

τ →∞

2

=0

(2.25)

provided that 〈Y(t)〉 = 0. Thus, we expect a general time-correlation function to be a decaying function of time, with an initial value given by the variance of Y. A reasonable choice might be the function: G(τ ) = G(0)exp(−|| τ /τ c )

(2.26)

The symbol τc is called correlation time. In Chapter 6, we will use a simple model for the random walk, depicted in Figure 2.1, to demonstrate that our choice of time-correlation function of the form given in Equation (2.26) can indeed be obtained for the spherical harmonics of the angles specifying molecular orientation in a liquid. The correlation time has many possible simple interpretations: it is a measure of the time scale of oscillations of the random process or a measure of the persistence of the correlation between values of Y(t) at different points in time. The two correlation functions in Figure 2.3 can thus be identified as being characterized by different correlation times. For the case of molecular reorientation in a liquid, we can treat τc as an average time for a molecular axis to change its direction by one radian. The quantities of prime interest in relaxation theory are spectral density functions, which are Fourier transforms of the tcf’s: ∞

J (ω ) =

∫ G(τ )exp(−iωτ )dτ

(2.27)

−∞

Because the concept of negative time is awkward, it may be more convenient to define the spectral densities as twice the one-sided Fourier transform of the time-correlation function: ∞

∫

J (ω ) = 2 G(τ )exp(−iωτ )dτ

(2.28)

0

According to the Wiener–Khinchin theorem in the theory of stochastic processes, the spectral density has a straightforward physical interpretation: it

Simple Relaxation Theory

29

−5 × 109 −3 × 109 −1 × 109 1 × 109 ω(rad s−1)

3 × 109

5 × 109

FIGURE 2.4 Spectral density function calculated with a correlation time τc = 2 ns.

is a measure of the distribution of the fluctuations of Y(t) among different frequencies. The spectral density associated with the exponentially decaying tcf is Lorentzian: J (ω ) = G(0)

2τ c 1 + ω 2τ c2

(2.29)

The shape of the Lorentzian spectral density is indicated in Figure 2.4.

2.3

Time-Dependent Perturbation Theory and Transition Probabilities in NMR Relaxation Theory

Very few problems in quantum mechanics can be solved exactly; therefore, techniques of approximation are needed. If we can somehow relate the problem at hand to a system with known solution, we can use this to get further. If the most important part of the system can be characterized by ˆ and with analogy with a known system, described by the Hamiltonian H 0 known solutions, we can focus on the small addition to the main part. In general, perturbation theory is based on the assumption that the Hamiltonian for the system under consideration can be expressed as a sum of the ˆ (usually independent of time), and a smaller term main, unperturbed part, H 0 ˆ are assumed to be known (or terms), called perturbation. The eigenstates of H 0 and the theory is applied to find how these are modified by the presence of the perturbation. The perturbation may be constant in time or time dependent. The approximation technique is referred to as perturbation theory. Quantum mechanical calculations of transition probabilities are based on the timedependent perturbation theory, in which the perturbation is assumed to be time dependent. The somewhat sketchy derivation here is based on the book of Carrington and MacLachlan.1 For a more complete and formal presentation, the reader is referred to the book of Atkins and Friedman (see further reading).

30

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

Consider a simple case of a two-level quantum system (energy levels Ea and Eb, eigenfunctions ψa and ψb), which is acted on by a Hamiltonian ˆ , and time-dependent perturbacontaining a sum of the main part, H 0 ˆ tion V (t) . We look for approximate solutions to the time-dependent Schrödinger equation, formulated in Equation (1.9), in the form:

ψ (t) = ca (t)ψ ae − iEat + cb (t)ψ b e − iEbt

(2.30)

Equation (2.30) is formally very similar to Equation (1.11a) and Equation (1.11b). The physical meaning of Equation (2.30) is that the perturbation does not change the nature of the problem in any fundamental way, i.e., the ˆ still provide a useful set of functions in which the approxeigenstates of H 0 ˆ =H ˆ + Vˆ (t) can be expanded. In NMR, this is never imate solutions to H 0 really a problem: the eigenstates to the Zeeman Hamiltonian form a complete set of functions in the spin space, i.e., any arbitrary spin function for I = 1/2 can be expressed exactly in the form of Equation (2.30). Assume vanishing diagonal elements of Vˆ (t) , i.e., the Hamiltonian represented by the matrix:  Ea Vab (t)    Eb  Vba (t)

(2.31)

with the matrix elements Vab (t) = 〈 a|Vˆ (t)|b 〉 = Vba* (t) (the perturbation is a hermitian operator). It is easy to show that the coefficients fulfill the equations: i

dca (t) = Vab (t)e i ( Ea − Eb )t cb (t) = Vab (t)e iω abt cb (t) dt

(2.32a)

i

dcb (t) = Vba (t)e i ( Eb − Ea )t ca (t) = Vba (t)e − iω abt ca (t) dt

(2.32b)

where ωab = (Ea – Eb) and all energy-related quantities are in angular frequency units. Note that the equations for the coefficients are coupled. Suppose that the system is in the state a at t = 0 (ca(0) = 1, cb(0) = 0) and that the perturbation is weak, i.e., the coefficients change slowly. We can then obtain an approximate expression for the coefficient cb(t) as: t

∫

cb (t) = −i Vba (t′)e − iω abt′ dt′

(2.33a)

0

t

∫

c (t) = i Vba* (t′)e iω abt′ dt′ * b

0

(2.33b)

Simple Relaxation Theory

31

The probability that the system is in state b is expressed by |cb(t)|2 = cb(t) cb*(t). The transition probability per unit time is the rate of change of this quantity: Wab =

2 d dc (t) dc * (t) cb (t) = b cb* (t) + cb (t) b dt dt dt

(2.34)

The expression for dcb(t)/dt in Equation (2.32b) can be rearranged slightly to: dcb (t) = −ie − iω abtVba (t) dt

(2.35)

We can use this result together with Equation (2.34) to obtain: t

Wab = e

− iω abt

∫

Vba (t) Vba* (t ′)e iω abt′ dt ′ + c.c. 0

(2.36)

t

=

∫ V (t)V (t′)e ba

* ba

iω ab ( t′− t )

dt ′ + c.c.

0

where c.c. denotes complex conjugate. When discussing relaxation-related transitions in a liquid, we need to know an average behavior of a large number of systems with stochastic Vˆ (t) , i.e., with the perturbation varying from one member in the spin ensemble to another in a random way. To describe the average behavior in an ensemble of spins, we make the variable substitution τ = t′ − t and take at the same time the ensemble average of Equation (2.36): t

Wab =

∫ V (t)V (t + τ ) e * ba

ba

iω abτ

dτ + c.c.

(2.37)

0

We can note, in passing, that according to the ergodic hypothesis of statistical mechanics, the ensemble average is equivalent to the time average taken for a single system over a long time. We note that the integral in Equation (2.37) contains the tcf of Vba(t), 〈Vba(t)V *ba(t + τ)〉. Making use of the properties of tcf’s we can, following Equation (2.23), write G*ba(τ) = 〈Vba(t)V *ba(t + τ)〉 and we obtain: t

∫

Wab = Gba (τ )e iω abτ dτ −t

(2.38)

32

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

Let us now assume that we wish to study the transition probabilities on a time scale much larger than the correlation time characterizing the decay of the time-correlation function, t >> τc. Under this condition, G(τ) vanishes at ±t and the integration limits can be extended to ±∞. Using Equation (2.23) once more, we obtain for the transition probability: ∞

Wab =

∫

∞

Gba (τ )e iω abτ dτ =

−∞

∫ G (τ )e ba

− iω abτ

−∞

∞

dτ (2.39)

∫

= 2 Gba (τ )e − iω abτ dτ = Jba (ω ab ) 0

Thus, we obtain the very important result that the transition probability, induced by a randomly fluctuating interaction, is equal to the spectral density of the random perturbation, evaluated at the frequency corresponding to the relevant energy level spacing. We introduce here a notation convention to be followed throughout this book. We use the symbol J(ω) for a spectral density of a purely classical random function; the script symbol with appropriate indices, here Jba(ω), denotes a spectral density for matrix elements of a stochastic operator. Through the Wiener–Khinchin theorem, the transition probability is therefore connected to the power available at the transition frequency. This tells us that the relaxation processes are in a sense related to the Einstein formulation of transition probabilities for stimulated absorption and emission of radiation; those are also proportional to the power available at the transition frequency. In passing, we can note that the spontaneous emission processes, which in principle could contribute to depopulating the upper spin state, can be completely neglected in NMR because of the low frequencies (energy differences) involved. For a more complete discussion of Einstein transition probabilities, consult the book by Atkins and Friedman (further reading).

2.4

Predictions of the Simple Model

We are now in a position to examine what the results mean for the relaxation rates. Combining Equation (2.3) for the spin-lattice relaxation rate with Equation (2.39) for the transition probability and Equation (2.29) for the spectral density, we arrive at a simple expression for T1: T1−1 = 2G(0)

2τ c 1 + ω 20 τ c2

(2.40)

Simple Relaxation Theory

33

Here, the frequency corresponding to the difference in energy levels according to the previous section is denoted by ω0 and is known as the Larmor, or resonance, frequency introduced in Chapter 1. According to Equation (2.24), G(0) is the variance or the mean-square amplitude of the interaction leading to relaxation. The average value of the interaction is assumed to be zero. Let us assume a simple (and physically not quite realistic) relaxation mechanism, which we can call the randomly reorienting field. We thus assume that the spins are subject to a local magnetic field (a vector) with a constant magnitude b, whose direction with respect to the large external field B0 (or to the laboratory z-direction) fluctuates randomly in small angular steps in agreement with Figure 2.1. This model is a simplified version of the effects of dipolar local fields, discussed in detail in Chapter 3. The model is also related to the fluctuating random field description as used in books by Slichter, Canet, and Levitt (see further reading). The mean-square amplitude of the fluctuating Zeeman interaction is then G(0) = γ 2Ib2 and we obtain: T1−1 = 4γ I2b 2

τc 1 + ω 02τ c2

(2.41)

Magnetic field has the units of tesla (T) and γ I has the units of rad s–1 T–1. Thus, the square of the interaction strength expression in front of the Lorentzian has the units rad2 s–2. As mentioned in Section 1.3, the radians are tacitly omitted, which leads to expression of the correlation time and the relaxation time in seconds. In full analogy with this case, in all other expressions in relaxation theory the interaction strength is expressed in radians per second, the correlation time in seconds, and the relaxation rate in s–1. The meaning of the correlation time in Equation (2.41) requires some reflection. The Zeeman interaction between the z-component of the nuclear magnetic moment and the local field can be written: Vˆ = −γ I Iˆzbz = −γ I Iˆzb cos θ

(2.42)

where we have used the fact that if the local field vector is oriented at angle θ with respect to the laboratory z-axis, then its z-component can be expressed as bcosθ. The angle θ and thus its cosine are, according to the model, a random function. The correlation time is related to the rate of decay of the correlation between cosθ(t) and cosθ(t + τ). For future reference, we note that, except for a normalization constant, cosθ is identical to the rank-1 spherical harmonics function, Y1,0. More about spherical harmonics can be found in the books by Atkins and Friedman and Brink and Satchler (see further reading). We will come to correlation functions for spherical harmonics in Chapter 6. It will be demonstrated there that the correlation function of Y1,0 is indeed

34

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

1/T1 (s−1)

20 15 10 5 0 100

102

104

106 ω0(rad s−1)

108

1010

1012

FIGURE 2.5 1/T1 calculated as a function of Larmor frequency for two correlation times, τc =0.2 ns (lower curve) and τc = 2 ns, using Equation (2.41). The value γI2b2 = 2.15 × 109 s–2 was used.

an exponential decay as in Equation (2.26). We will call the correlation time in Equation (2.41) rotational correlation time (to be strict, we should really add “for a rank-1 spherical harmonics”) related to the reorientation of the molecule carrying the spins. Indeed, using the results to be obtained in Section 6.1, we should insert a factor 1/4π into the right-hand-side of Equation (2.41). This does not matter too much because we are only interested here in following the variation of the T1–1 with certain physical quantities and not in its absolute magnitude. We can use the simple form of Equation (2.41) to predict the dependence of the spin-lattice relaxation rate on the strength of the external magnetic field, B0 (through the Larmor frequency), and on the molecular size and temperature through the correlation time. Let us begin by looking at the dependence of T1–1 on the magnetic field at a constant value of the correlation time. A plot of the relaxation rate vs. ω0 is shown in Figure 2.5 for two physically reasonable values of τc: 200 ps and 2 ns. According to the plot, the relaxation rate is expected to be constant at low field and decrease dramatically in the vicinity of the condition ω0τc = 1. The rapid reduction of the relaxation rate is sometimes called dispersion. The curves in Figure 2.5 map exactly the plot of the spectral density vs. frequency, as shown in Figure 2.4 (note the logarithmic scale on the horizontal axis in Figure 2.5), which of course reflects the fact that the two quantities are proportional to each other for the simple case at hand. The flat region of the plot of T1–1 against Larmor frequency, i.e., the region where the relaxation rate is independent of the magnetic field, is called the extreme narrowing region. In quantitative terms, the extreme narrowing range corresponds to the condition ω02τc2 << 1. The extreme narrowing regime extends up to a certain value of the Larmor frequency (magnetic field); the range is smaller for a longer correlation time. One more observation we can make from Figure 2.5 is that, for a given correlation time, the nuclei with lower magnetogyric ratio will come out of extreme narrowing regime at a higher field. Thus, protons with their high magnetogyric ratio come out of extreme narrowing regime at a lower field than 13C or 15N (compare Table 1.1).

Simple Relaxation Theory

35

Besides the Larmor frequency or the magnetic field, the second interesting variable in Equation (2.41) is the rotational correlation time. We will arrive at a stringent definition of this quantity later, but we wish to state its dependence on molecular size, solution viscosity, and temperature here. Using hydrodynamic arguments for a spherical particle with the hydrodynamic radius a and volume V = 4πa3/3, reorienting in a viscous medium, one can derive the Stokes–Einstein–Debye (SED) relation, introduced in the NMR context in the classical paper by Bloembergen, Purcell, and Pound (BPP)2:

τ c (l = 1) =

4πηa3 3Vη = k BT k BT

(2.43a)

Here, η is the solution viscosity (in the units kg s–1 m–1) and V is the volume of the molecule. Clearly, the volume or the a3 factor in the numerator indicates that the rotational correlation time is expected to increase with the molecular size. Indeed, “rule of thumb” relations for aqueous protein solutions relate the correlation time to the molecular weight. We should notice that Equation (2.43a) applies for the correlation time for l = 1 spherical harmonics. Several of the physically more realistic relaxation mechanisms (e.g., the dipole–dipole relaxation) depend on the rotational correlation time for l = 2 spherical harmonics. In that case, Equation (2.43a) is modified to:

τ c (l = 2) =

4πηa3 Vη = k BT 3 k BT

(2.43b)

The correlation time depends on temperature in two ways. First, the viscosity is strongly temperature dependent, which is commonly described by an Arrhenius-type expression (proposed for the first time almost 100 years ago by deGuzman3):

η = η0 exp(Eaη/kBT )

(2.44)

where Eaη is the activation energy for viscous flow and η0 is a constant without too much physical significance. Second, the 1/T dependence originates from the presence of temperature in the denominator of Equation (2.43a) and Equation (2.43b). The exponential factor overruns the 1/T dependence, and it is common to express the temperature dependence of the correlation time by an analogous Arrhenius-type expression:

τ c = τ 0 exp(Eaτ/kBT )

(2.45)

where the activation energy Eaτ is related to the barrier hindering the reorientation process and not necessarily the same as Eaη. The symbol τ0 is a constant, again without too much physical significance.

36

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

100.00

T1 (s)

10.00

1.00

0.10

10-12

10-11

10-10

10-9

10-8

10-7

τc (s)

FIGURE 2.6 Plot of T1 vs. correlation time, τc. The plot was calculated using Equation (2.41), assuming γI2b2 = 1.13 × 109 for a proton at a magnetic field strength of 9.4 T (corresponding to a 1H resonance frequency of 400 MHz).

A plot of T1 vs. correlation time for a given magnetic field B0 or Larmor frequency ω0 is shown in Figure 2.6. When molecular motions are rapid so that τc2ω02 << 1, the extreme narrowing region prevails, the frequency dependence in the denominators of Equation (2.41) and Equation (2.29) vanishes, and we get the result that T1–1 is proportional to the correlation time (the lefthand side of the diagram in Figure 2.6). Turning to longer correlation times, we can see in the figure that the relaxation is most efficient (the T1–1 is largest or the T1 shortest) when τc = 1/ω0. At a typical NMR field of 9.4 T (400 MHz proton resonance frequency or | ω0 | = 2π ⋅ 400 ⋅ 106 rad s–1), this happens for protons at τc = 400 ps, which is a typical rotational correlation time for a medium size organic molecule in an aqueous solution at room temperature. When we study even larger molecules or solutions of high viscosity in which the motions are more sluggish, the opposite condition, τc2ω02 >> 1, applies and the relaxation rate becomes inversely proportional to the correlation time (the right-hand side of the diagram in Figure 2.6). According to what we stated earlier concerning the dependence of the correlation time on molecular size and temperature, the horizontal axis in Figure 2.6 can be thought of as corresponding to the molecular weight. Similarly, we can think of temperature increasing to the left in the diagram. The case of the spin–spin relaxation time (transverse relaxation time) is a little more complicated. It turns out that the corresponding rate for the simple example at hand is proportional to the sum of spectral densities at frequencies zero and ω0:  τc  T2−1 = 2γ I2b 2  + τc 2 2  1 + ω 0τ c 

(2.46)

Simple Relaxation Theory

37

Relaxation rate (s−1)

100.00 10.00

1/T2

1.00 0.10 0.01 10−12 10−11 10−10 10−9 τc(s)

1/T1

10−8

10−7

FIGURE 2.7 Plot of 1/T1 and 1/T2 relaxation rates vs. correlation time, τc. The plot was calculated using Equation (2.41) and Equation (2.46), assuming γI2b2 = 1.13 × 109 for a proton at a magnetic field strength of 9.4 T (corresponding to a 1H resonance frequency of 400 MHz).

The relation between T1–1 = R1 and T2–1 = R2 as a function of the correlation time is summarized in Figure 2.7. We can see that the two relaxation rates are equal when the extreme narrowing conditions hold and that R2, as opposed to R1, continues to increase (the NMR signal becomes broader; compare to Figure 1.4) with the increasing correlation time. Already at this stage, we can see that very important results are obtained from the relaxation rate parameters. To summarize, we have seen that relaxation can provide detailed information about molecular motion and size through the field dependence of the relaxation providing the rotational correlation time. This will be discussed in detail in Chapter 11 and Chapter 12, in which applications of relaxation measurements are discussed.

2.5

Finite-Temperature Corrections to the Simple Model

Before leaving our simple model, we wish to discuss one principally important complication that occurs also in physically more realistic situations. The issue is the assumed equality of the transition probabilities, Wαβ = Wβα = WI, which was introduced at the beginning of this chapter. If this indeed were true, i.e., if the transition probabilities up and down in the two-level system were equal, then the spin system would evolve towards a situation with equal populations of the two levels. According to the Boltzmann distribution, this would correspond to infinite temperature. If we want our spin system to evolve towards thermal equilibrium at a finite temperature, we need to introduce small correction terms to our transition probabilities. For

38

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

a spin with positive magnetogyric ratio, implying that the α state is lower in energy than the β state, we need to write: Wα β = WI (1 − bI )

(2.47a)

Wβα = WI (1 + bI )

(2.47b)

The symbol WI is an average transition probability, given in our two-level model by Equation (2.39), and bI is the Boltzmann factor introduced in Equation (1.19a) and Equation (1.19b). We remind the reader that the Boltzmann factors in NMR are so tiny that the practical implications of the correction terms are small. However, when we use the assumption of Equations (2.47), the flow of populations in both directions becomes equal: nαeqWαβ = nβeqWβα

(2.48)

as indeed it should be at the thermal equilibrium. In the way in which Equation (2.47a) and Equation (2.47b) are formulated, the thermal correction terms can be considered as inserted ad hoc, without any more profound motivation than that they lead to the correct results of Equation (2.48). It is possible to derive Equation (2.47a) and Equation (2.47b) starting from the fundamental physical principles, if one assumes that the environment of spins (the lattice) is also of quantum mechanical nature and recognizes that simultaneous transitions in the spin system and the lattice should be considered. The details of that problem are, however, beyond the scope of this book and we recommend consulting the book of Abragam (see further reading).

References 1. Carrington, A. and McLachlan, A.D. Introduction to Magnetic Resonance (New York: Harper and Row), 1967. 2. Bloembergen, N., Purcell, E.M., and Pound, R.V. Relaxation effects in nuclear magnetic resonance absorption, Phys. Rev., 73, 679–712, 1948. 3. de Guzman, J. Relation between fluidity and heat of fusion, Anales soc. espan. fis. quim, 11, 353–362, 1913.

3 Relaxation through Dipolar Interactions

CONTENTS 3.1 The Nature of the Dipolar Interaction .....................................................40 3.2 The Solomon Relaxation Theory for a Two-Spin System .....................45 3.2.1 The Dipolar Alphabet.....................................................................46 3.2.2 The Solomon Equations .................................................................50 3.3 The Nuclear Overhauser Enhancement...................................................53 3.4 Carbon-13 Relaxation..................................................................................56 3.5 Intermolecular Dipolar Relaxation ...........................................................59 References ............................................................................................... 63

Until now, we have only considered relaxation through a simple approach, using a model relaxation mechanism. In this chapter we will deal with the dipole–dipole (DD) relaxation, which has proven to be one of the most important sources for obtaining molecular dynamics information. The discussion on this relaxation mechanism is based on the Solomon equations, which, in turn, are based on rates of transitions between spin states. For many spin 1/2 nuclei, the dipole–dipole interaction between nuclear magnetic moments of spins in spatial vicinity of each other is the most important relaxation mechanism. This was recognized in the seminal work by Bloembergen, Purcell, and Pound (BPP) from 1948 (mentioned in the previous chapter), which created the ground for all subsequent development of relaxation theory. What was not correct in the BPP paper was the assumption that one of the spins acts as a source of a kind of random field for the other spin, on which the interest was concentrated. As shown by Solomon in 1955,1 the dipolar relaxation must be dealt with in terms of a four-level system with two mutually interacting spins. In this chapter, we go through the Solomon theory for the dipolar spin-lattice relaxation and its consequences. Good description of dipolar interaction and dipolar relaxation can be found in the books by Abragam and by Hennel and Klinowski (see further reading in the Preface).

39

40

3.1

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

The Nature of the Dipolar Interaction

Consider a situation in which we have two nuclear magnetic moments or magnetic dipoles, µ1 and µ2, close to each other in space. Each of the magnetic dipoles generates around itself a local magnetic field. The local magnetic field is a vector field, i.e., it is at every point characterized by a magnitude and a direction (compare Figure 3.1). The form of the vector field generated by a magnetic dipole is rather complicated. Let us consider the field created by the dipole µ2; as discussed by Atkins and Friedman (see further reading), the magnetic field at the point r (with respect to µ2 at the origin) can be expressed by: Bloc ( µ2 ) = −

 µ0  rr µ − 3 2 ⋅ µ2  3  2 4π r  r 

(3.1)

where µ0 is the permeability of vacuum µ0/4π is in the SI units equal to 10–7 Js2C–2 m–1 or T2J–1m3 r (a scalar quantity) is the distance from the origin r is a vector with Cartesian components x, y, z rr is a tensor, an outer product of two vectors A tensor has the property that when it acts on a vector, the result is another vector with, in general, different magnitude and orientation. A Cartesian z

x

FIGURE 3.1 The local magnetic field from a magnetic point dipole along the positive z-axis at origin.

Relaxation through Dipolar Interactions

41

tensor can be represented by a 3 × 3 matrix. An outer product of two vectors, a and b, with Cartesian components ax, ay, az and bx, by, bz, respectively, can be represented by a matrix with elements:  ax    ab =  ay  bx  az 

(

by

 axbx  bz =  aybx  azbx 

)

axby ay b y azby

axbz   aybz  azbz 

(3.2a)

Thus, the rr tensor can be expressed as:  xx xy xz  x 2 xy xz     rr =  yx yy yz =  yx y 2 yz  zx zy zz   zx zy z 2   

(3.2b)

The operation of a tensor on a vector can then be represented as a multiplication of that matrix with a vector. The second term in the expression in Equation (3.1) is thus a vector whose magnitude and direction depend on the point in which we are interested (as an exercise, the reader is advised to derive the z component of the local field at r). The local magnetic field created by the dipole µ2 interacts with µ1 and, in the same way, the local field of µ1 interacts with µ2. The classical dipole–dipole interaction energy, EDD, is: EDD =

µ0 4π r 3

  rr  1 ⋅ 2 − 31 ⋅ 2 ⋅ 2  r

(3.3)

where r = xi + yj + zk (i, j, and k are the unit vector along the three Cartesian axes) now denotes the vector connecting the two dipoles. The quantum-mechanical counterpart of the classical dipole–dipole energy expression is the dipole–dipole Hamiltonian, which we obtain replacing the magnetic dipoles by γ I Iˆ and γ SSˆ , corresponding to the two spins denoted as I and S:   ˆ = − µ0γ Iγ S 3Iˆ ⋅ rr ⋅ Sˆ − Iˆ ⋅ Sˆ = b Iˆ ⋅ D ⋅ Sˆ H  DD IS 3  2 r 4π r  

(3.4)

If the distance between spins (the length of the r vector) is constant and is equal to rIS, then the quantity bIS, given by: bIS = −

µ0γ Iγ S 4π rIS3

(3.5)

42

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

is also a constant, denoted as the dipole–dipole coupling constant. Note that the Hamiltonian in Equation (3.4) is in the same units as the dipole–dipole coupling constant, i.e., in the angular frequency units (rad s–1). In analogy with Equation (3.2a) and Equation (3.2b), the dipolar tensor D can be represented by a 3 × 3 matrix:  3x 2 3 xy 3 xz   2 −1  2 rIS rIS2   rIS  3 xy 3y2 3 yz  D= −1  2 2 rIS2  rIS  rIS  3 xz  3 yz 3z2 − 1  2 2 2 rIS rIS  rIS   3 sin 2 θ cos 2 φ − 1  =  3 sin 2 θ cos φ sin φ  3 sin θ cos θ cos φ 

(3.6)

3 sin 2 θ cos φ sin φ 3 sin θ sin φ − 1 2

2

3 sin θ cos θ sin φ

3 sin θ cos θ cos φ   3 sin θ cos θ sin φ  3 cos 2 θ − 1 

In the second line of Equation (3.6), we express the elements of the Cartesian tensor D in spherical polar coordinates (r, θ, φ), using standard relations: x = rsinθcosφ, y = rsinθsinφ, z = rcosθ. When working with the Hamiltonian of Equation (3.4) or with the dipolar tensor of Equation (3.6), we must be careful with the choice of the coordinate system. The tensor takes on a particularly simple form in the coordinate frame where the z-axis coincides with the vector r. We then have x = y = 0 (θ = φ = 0) and z = rIS. All the off-diagonal elements of D vanish and the diagonal elements become: –1, –1, and 2. In the presence of a strong magnetic field, the main interaction the spins sense is the Zeeman interaction. Therefore, the natural choice of the coordinate frame for the spin operators is the laboratory frame, where the z-direction is defined by the magnetic field. The orientation of the vector r should thus also be specified in the laboratory frame (compare Figure 3.2). Before turning to the role that the dipole–dipole interaction has in relaxation, let us have a quick look at how it demonstrates itself in solid crystals and powders. Begin by assuming that the spins I and S are different species (e.g., a carbon-13 and a proton) and that we deal with isolated spin pairs, i.e., that each spin interacts with one and only one other spin. In a high magnetic field, where the Zeeman interactions dominate, we can make use of the secular approximation. This approximation means that we only retain the terms in HDD that commute with the Zeeman Hamiltonian, which for the case of two spins can be written, in analogy with Equation (1.6), as: ˆ = −(γ Iˆ + γ Sˆ )B = ω Iˆ + ω Sˆ H z I z S z 0 I z S z

(3.7)

Relaxation through Dipolar Interactions

43

FIGURE 3.2 The orientation of IS-spin vector, r, in the laboratory coordinate system, defined by the angles θ and φ.

The dipole–dipole Hamiltonian then attains a simplified form: ˆ sec = b (3 cos 2 θ − 1)Iˆ Sˆ H DD IS z z

(3.8)

Here, θ is the angle between the axis joining the two dipolar-coupled spins (fixed in the crystal) and the magnetic field direction, assumed to be the laboratory frame z-direction. The angle θ between the laboratory z-direction and the internuclear axis, the r vector, is defined in Figure 3.2. The results of the presence of the dipolar Hamiltonian is that the I- and S-spin resonance lines split into doublets, in the same way as they do in the presence of the scalar spin–spin coupling (indirect spin–spin coupling or J-coupling). The splitting, which for the indirectly coupled spins is equal to 2πJ (note the angular frequency units), is here bIS(3cos2 θ – 1). The splitting thus depends on the orientation of the crystal in the magnet. Rotating the crystal in the magnet in such a way that the θ angle varies between 0 and π results in the splitting changing as in Figure 3.3. When comparing the dipole–dipole coupling with the J-coupling, we should be aware of the interaction strengths involved: the J-coupling for a directly bonded carbon-13–proton pair is typically about 800 rad s–1 (130 Hz), and the corresponding dipolar coupling constant, bIS, is about 140,000 rad s–1 (22 kHz). Let us then turn to a solid powder, i.e., a system of a large number of microcrystallites with random orientation with respect to each other and to the magnetic field. Every orientation with respect to the laboratory frame z-axis contributes to the spectrum with a particular splitting according to Figure 3.3; the total spectrum is a superposition of all the doublets. The intensities of lines with various splittings depend on probabilities of the corresponding θ angles. The assumption that all orientations of the microcrystallites are

44

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

DD splitting (kHz)

60 40 20 0

1

−20

2

3

−40 θ(rad) FIGURE 3.3 The variation of the dipolar splitting for a 1H-13C spin pair, with bCH = 144⋅103 rad s–1 (bCH/2π = 22.9 kHz). The value for bCH was obtained assuming an internuclear distance of rCH =1.09 Å.

equally probable does not imply that all the θ angles are equally probable. Using a “hand-waving” argument, we can imagine that the perpendicular orientations, θ = π/2 are much more numerous, corresponding to all possible vector orientations in the xy-plane, and can be realized much more often than the parallel orientations with only two possibilities, θ = 0 or π. In mathematical terms, we need to think about the angular volume element being sinθ dθ dφ, where φ is the angle specifying the orientation of the projection of the IS axis on the xy-plane. The powder spectrum of the I-spin, dipole–dipole coupled to the S-spin is shown in Figure 3.4. The pattern is called powder pattern or Pake doublet, after the scientist who reported it for the first time.2 The splitting between the maxima in the powder pattern is equal to | bIS |, which corresponds to θ = π/2. We can note, before leaving the solids, that the dipolar splitting vanishes if 3cos2 θ – 1 = 0. This happens when θ = arccos(1/ 3 ) = 54.74°. This angle is called

bIs/2π

−40

−20

0 Frequency(kHz)

20

40

FIGURE 3.4 Powder spectrum, or Pake pattern, of the I-spin, dipole–dipole coupled the S-spin with bIS = 144⋅103 rad s–1 (bIS/2π = 22.9 kHz).

Relaxation through Dipolar Interactions

45

the magic angle. An important technique in solid-state NMR is to spin the powder sample around an axis oriented at the magic angle with respect to the magnetic field. If the spinning motion imposed by the experimenter is sufficiently rapid, the dipole–dipole interaction (as well as other interactions with the same θ dependence) is “projected” on the spinning axis and the dipole–dipole interaction vanishes from the spectrum. The dipole–dipole splittings vanish also in isotropic liquids. This is the effect of motion here also, but now it is imposed by nature of the sample rather than the experimenter. Molecular tumbling in ordinary liquids changes the orientation of the IS-spin axis on a time scale that is very fast compared to the dipole–dipole couplings. The rotational correlation times are on the order of pico- or nanoseconds, corresponding to reorientation rate constants of 107 – 1011 rad s–1, which should be compared to the dipolar couplings on the order of 105 rad s–1 (compare to the preceding example). Thus, the dipole–dipole coupling in a liquid represents an average of bIS(3cos2 θ – 1) over the isotropic distribution of the orientations, which amounts to zero: 2π π

bIS ( ave) = bIS

∫ ∫ (3 cos θ − 1)p(θ , φ)sinθ dθ dφ 2

0 0

=

bIS 4π

(3.9)

2π π

∫ ∫ (3 cos θ − 1)sinθ dθ dφ = 0 2

0 0

The symbol p(θ,φ) is the probability density for the angles θ and φ. The probability density is uniform and equal to 1/4π because of the normalization condition. The reader is encouraged to perform the integration in the second line of Equation (3.9). Thus, the dipole–dipole interaction does not give splittings in isotropic liquids, and the fine structure of NMR spectra in such systems is determined only by the J-couplings. This does not mean, however, that the dipole–dipole interaction cannot be effective as a relaxation mechanism. The ensemble averaged dipole–dipole ˆ (t)〉 = 0 (we Hamiltonian (Equation 3.4) vanishes at every point in time, 〈 H DD have explicitly shown that this is true for the secular part, Equation 3.8, but the same applies to the other terms). This does not imply that the ensemble averages ˆ (t)H ˆ (t + τ )〉 , relevant in the relaxation theory, need to vanish. of the type 〈 H DD DD

3.2

The Solomon Relaxation Theory for a Two-Spin System

Let us consider a liquid consisting of molecules containing two nuclei with spins I and S characterized by the spin quantum number 1/2. We assume that they are distinguishable, i.e., belong to different nuclear

46

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications βIβS

W1S

W1I

W2

βIαS

W0

W1I

αIβS

W1S

αIαS FIGURE 3.5 Energy-level diagram associated with the IS-spin system.

species or have different chemical shifts. We assume that there are no J-couplings, but allow for the dipole–dipole interaction between the spins. The system of such two spins is characterized by four Zeeman energy levels and by a set of transition probabilities among the levels (compare Figure 3.5). 3.2.1

The Dipolar Alphabet

The Zeeman Hamiltonian of Equation (3.7) can be treated as the main Hamiltonian, with the eigenstates αα, αβ, βα, and ββ (the first symbol indicates the state of the I-spin, the second that of the S-spin) and the dipole–dipole Hamiltonian of Equation (3.4) as a perturbation. We shall find it useful to express the dipole–dipole Hamiltonian as a sum of six terms: ˆ + Bˆ + Cˆ + D ˆ DD = b [ A ˆ + Eˆ + Fˆ ] H IS IS

(3.10)

The dipole–dipole coupling constant is for the time being assumed to be constant in time. This amounts to treating the molecules carrying the two spins as rigid objects. We will return to the intermolecular dipolar relaxation in the final section of this chapter and to the intramolecular dipolar relaxation in the case of flexible molecules in Chapter 6. The form of the dipole-dipole Hamiltonian as given in Equation (3.10) is often referred to as the “dipolar alphabet.” All the terms in Equation (3.10) consist of products of spin operators and angular terms proportional to the

Relaxation through Dipolar Interactions

47

rank-2 spherical harmonics: ˆ = Iˆ Sˆ (3 cos 2 θ − 1) A z z

(3.11a)

Bˆ = − 14  Iˆ+Sˆ − + Iˆ−Sˆ +  (3 cos 2 θ − 1)  

(3.11b)

Cˆ = 23  Iˆ+Sˆ z + IˆzSˆ +  sin θ cos θ exp(−iφ )  

(3.11c)

ˆ = 3  Iˆ Sˆ + Iˆ Sˆ  sin θ cos θ exp(iφ ) D z − 2  − z

(3.11d)

Eˆ = 34 Iˆ+Sˆ + sin 2 θ exp(−2iφ )

(3.11e)

Fˆ = 34 Iˆ−Sˆ − sin 2 θ exp(2iφ )

(3.11f)

One can easily derive Equation (3.10) and Equations (3.11) from Equation (3.4) and Equation (3.6), using the relation exp(iα) = cosα + i sinα and the stepup and step-down operators, Iˆ+ , Sˆ + and Iˆ− , Sˆ − , respectively, defined as: Iˆ± = Iˆx ± iIˆy

(3.12)

We recognize that the term  is identical to the secular part of the dipolar Hamiltonian, as given in Equation (3.8). When the step operators operate on the|I,m〉 kets, they produce other kets, with the m quantum number increased (Î+) respectively decreased (Ζ) by ˆ , in the one. Thus, the terms Bˆ − Fˆ correspond to the nonsecular part of H DD sense that the spin operators do not commute with the Zeeman Hamiltonian. ˆ are similar to each other: the spin We can also note that the terms Cˆ and D operators contain products of a step operator and a z-component operator and the angular parts are complex conjugates of each other. In the same sense, Ê and Fˆ terms are related to each other. One can make use of these relations and formulate the DD Hamiltonian in terms of an interaction strength constant, certain combinations of spin operators, called rank-2 irreducible spherical tensor operators, Tˆ2 ,m , and suitably normalized spherical harmonics Y2,m: 2

ˆ DD = ξ DD H IS IS

∑ (−1) Y m

Tˆ

2 ,m 2 ,− m

(3.13)

m=−2

The interaction strength constant, ξISDD = bIS 24π 5 , plays the same role as the dipole–dipole coupling constant; it is introduced in Equation (3.13) instead of bIS so that the other factors can have appropriate normalization (see Section 4.2). Here, Tˆ2 ,m represents the time-independent tensor operator acting on spin variables only, corresponding to the products of spin operators in Equations (3.11).

48

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

The formulations of Equations (3.11) and Equation (3.13) are fully equivalent; the choice of one or another is a matter of convenience. In the context of the Solomon theory, we choose to work in terms of the dipolar alphabet and will use the formulation of Equation (3.13) later. The reader interested in general aspects of various formulations of Hamiltonians for NMR can find it useful to consult the early paper by Spiess3 and the extensive series of reviews by Smith and coworkers.4–6 Because of the angular (θ, φ) dependences, the dipole–dipole interaction is influenced (and averaged to zero on a sufficiently long time scale) by molecular tumbling. Taking this motional variation into considerations, it is not particularly difficult to use time-dependent perturbation theory to derive transition probabilities between the pairs of levels in Figure 3.5. The transition probabilities are obtained in a way similar to the case discussed in Chapter 2. They are related to time-correlation functions ˆ DD (t))ij ( H ˆ DD (t + τ ))ij 〉 where ( H ˆ (t)) denotes the ij matrix element of the 〈( H DD ij dipolar Hamiltonian and the corresponding spectral density functions, which reflect the intensity of the local magnetic fields fluctuating at the frequencies corresponding to the energy level differences in Figure 3.5. ˆ − Fˆ ), a total of 36 ˆ (t) is expressed as a sum of six terms ( A Because the H DD ˆ ˆ (t + τ )) . terms will in principle be present in the product ( HDD (t))ij ( H DD ij ˆ Because all the terms in HDD (t) are products of time-independent spin operators and time-dependent angular functions, the calculation of transition probabilities involves two steps. First, we need to calculate the matrix elements of the time-independent spin operators in the basis of the eigenkets to the Zeeman Hamiltonian. Doing that, we find that different pairs of levels are connected by matrix elements of different operators. In other words, different transitions are induced by different terms of the dipolar alphabet. The relations between pairs of levels and the operators connecting them are summarized in Table 3.1. TABLE 3.1 Mixing of IS-Spin States by Various Terms of the Dipolar Hamiltonian Spin state

ββ

αβ

βα

αα

ββ

ˆ A

ˆ D

ˆ D

Fˆ

αβ

Cˆ

ˆ A

Bˆ

ˆ D

βα

Cˆ

Bˆ

ˆ A

ˆ D

αα

Eˆ

Cˆ

Cˆ

ˆ A

Source: adapted from Hennel, J.W. and Klinowski, J., Fundamentals of Nuclear Magnetic Resonance (Harlow: Longman), 1993.

Relaxation through Dipolar Interactions

49

Only the  -symbol (secular terms) occurs on the diagonal. The occurrence of the other terms can be illustrated by examining one of the matrix elements ˆ |ββ 〉 . Operation of the term  in the dipolar alphabet — for instance, 〈αα |H DD of Equations (3.11) on the | ββ 〉 ket leaves it unchanged and the matrix element vanishes because of the orthogonality of the spin functions. The terms ˆ , and Fˆ ) operating on |ββ〉 yield zero containing a step-down operator ( Bˆ , D because the ket with m = –1/2 and I = 1/2 cannot have its m-quantum number reduced further. The term Cˆ operating on |ββ 〉 yields a linear combination of | αβ 〉 and | βα〉, both of which are orthogonal to the | αα〉, and the matrix element vanishes for the same reason as that of the Â-operator. Thus, the only term left is Eˆ ; the spin operator Iˆ+Sˆ + acting on | ββ 〉 raises the m-quantum number of both spins and thus yields a ket proportional to | αα〉. Taking the scalar product with | αα〉 results in a nonzero matrix element corresponding to the Eˆ -symbol in the lower-left corner of Table 3.1. In the second step of the calculations, we deal with the time-correlation functions of the angular functions in Equation (3.11a) through Equation (3.11f). These depend on the nature of random reorientations and thus on the model we choose to describe these motions. In Chapter 6, we will derive an expression for the time-correlation function for some simple models, including isotropic reorientation of a rigid body in small, diffusional steps. For the time being, we assume that the relevant time-correlation function for a normalized Y2,0 spherical harmonics:

Y2 ,0 =

5 (3 cos 2 θ − 1) 16π

(3.14)

is

Y2 ,0 (t)Y2 ,0 (t + τ ) =

1 exp(−τ/τ c ) 4π

(3.15)

That is, it has the same form as that of Equation (2.26). The corresponding spectral densities are again Lorentzian, similar to those of Equation (2.29):

J (ω ) =

1 2τ c 1 τc = 2 2 4π (1 + ω τ c ) 2π (1 + ω 2τ c2 )

(3.16)

Note that the factor 1/4π corresponding to G(0) is carried over from Equation (3.15) to Equation (3.16). The correlation time in Equation (3.15) and Equation (3.16) is similar, but not identical, to that introduced in Chapter 2. In the small-step rotational diffusion model, the correlation time for spherical

50

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

harmonics with a given l value is inversely proportional to l(l + 1). This relation will be derived in Chapter 6. Here, we only note that the correlation time relevant for the dipole–dipole relaxation (l = 2) is expected to be three times smaller than the correlation time for the randomly reorienting local field, l = 1. This relation was reflected in Chapter 2 (Equations 2.43). 3.2.2

The Solomon Equations

In analogy with the simple model for the spin-lattice relaxation, we can set up a set of equations describing the populations’ kinetics in the four-level system of Figure 3.5. For the αα level, we can write: dNαα eq = −(W1I + W1S + W2 )(nαα − nαα ) dt

(3.17)

+ W2 (nββ − n ) + W1I (nβα − n ) + W1S (nα β − n ) eq ββ

eq βα

eq αβ

For the other levels, analogous equations can be obtained. The relations between the population differences and expectation values of z-magnetization for the two spins are given by: Iˆz ∝ nI = (nαα − nβα ) + (nαβ − nββ )

(3.18a)

Sˆ z ∝ nS = (nαα − nαβ ) + (nβα − nββ )

(3.18b)

Using Equation (3.18a) and Equation (3.18b), we can move on from the population kinetics to the magnetization kinetics. Solomon has showed that the relaxation of the longitudinal magnetization components, proportional to the expectation values of Iˆz and Sˆ z operators, is related to the populations of the four levels and can be described by a set of two coupled equations: d Iˆz dt d Sˆ z dt

= −(W0 + 2W1I + W2 )( Iˆz − I zeq ) − (W2 − W0 )( Sˆ z − Szeq )

(3.19a)

= −(W2 − W0 )( Iˆz − I zeq ) − (W0 + 2W1S + W2 )( Sˆ z − Szeq )

(3.19b)

or d Iˆz dt d Sˆ z dt

= − ρI ( Iˆz − I zeq ) − σ IS ( Sˆ z − Szeq )

(3.20a)

= −σ IS ( Iˆz − I zeq ) − ρS ( Sˆ z − Szeq )

(3.20b)

Relaxation through Dipolar Interactions

51

or more compactly as:  d dt  

eq Iˆz   ρ σ IS   Iˆz − I z     = − I   σ IS ρS   Sˆ z − Szeq  Sˆ z 

(3.21)

Izeq and Szeq are the equilibrium longitudinal magnetizations for the two spins (originating from the equilibrium populations in Equation 3.17), and ρI and ρS are the corresponding decay rates (spin-lattice autorelaxation rates or, briefly, spin-lattice relaxation rates). The symbol σIS denotes the crossrelaxation rate. By comparing Equations (3.19) and Equations (3.20), we can make the following identifications:

ρI = W0 + 2W1I + W2

(3.22a)

ρS = W0 + 2W1S + W2

(3.22b)

σ IS = W2 − W0

(3.22c)

We can rewrite Equations (3.20) in an even more compact vector–matrix expression: d V = −R(V − V eq ) dt

(3.23)

V is a column vector with the expectation values of Iˆz and Sˆ z operators as the elements, and Veq is its thermal equilibrium counterpart. The matrix R in the right-hand side of Equation (3.21) and Equation (3.23) is called relaxation matrix. The general solutions of the Solomon equations for 〈 Iˆz 〉 or 〈Sˆ z 〉 are sums of two exponentials: Iˆz = v11 exp(− λ1t) + v12 exp(− λ2t)

(3.24a)

Sˆ z = v21 exp(− λ1t) + v22 exp(− λ2t)

(3.24b)

where λ1 and λ2 are eigenvalues of the relaxation matrix, i.e., the values obtained when the relaxation matrix is transformed into a form with nonzero elements only on the diagonal by replacing the original vector basis set 〈 Iˆz 〉, 〈Sˆ z 〉 by suitable linear combinations. The parameters vij depend on the linear combination coefficients and the initial conditions. The eigenvalues λ1 and λ2 would be equal to the spin-lattice autorelaxation rates, ρΙ and ρS , if the cross-relaxation rate, σIS, were equal to zero. If this were the case, the relaxation matrix would then be diagonal when expressed in terms of the original basis set. The cross-relaxation phenomenon is, however, in principle always present in dipolar-relaxed systems (except

52

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

for certain combinations of Larmor frequencies and correlation times, which result in cancellation of terms in the expression for σIS) and the longitudinal magnetizations in a two-spin system do not follow the Bloch equations. Solomon also demonstrated that, besides the odd situation when σIS = 0, the simple exponential relaxation behavior of the longitudinal magnetization is recovered under certain limiting conditions: • The two spins are identical (e.g., the two proton spins in a water molecule). In this case, only the sum of the two magnetizations is an observable quantity. Moreover, we have W1I = W1S = W1 and the spin-lattice relaxation rate is given by: T1−1 = 2(W1 + W2 )

(3.25)

The reader is encouraged to demonstrate that this indeed is the case. • One of the spins, say S, is characterized by another, faster relaxation mechanism. We can then say that the S-spin remains in thermal equilibrium on the time scale of the I-spin relaxation. This situation occurs, for example, in paramagnetic systems, where S is an electron spin. The spin-lattice relaxation rate for the I-spin is then given by:

ρI = T1−I1 = W0 + 2W1I + W2

(3.26a)

• One of the spins, say I, is saturated by an intense radiofrequency (r.f.) field at its resonance frequency. These are the conditions applying for carbon-13 (treated as the S-spin), for example, under broadband decoupling of protons (I-spins). The relaxation rate is also in this case given by an S-spin analogue of Equation (3.26a):

ρS = T1−S1 = W0 + 2W1S + W2

(3.26b)

and is called the spin-lattice relaxation rate. When the I-spin is saturated, 〈 Iˆz 〉 = 0 and the equation for the evolution of the 〈Sˆ z 〉 becomes: d Sˆ z dt

= − ρS ( Sˆ z − Szeq ) − σ IS (− I zeq )

(3.27)

The steady-state solution ( d 〈Sˆ z 〉/dt = 0 ) is given by: Sˆ z where η =

γ I σ IS γ SρS

steady − state

= Szeq +

 σ IS eq γ σ  I z = Szeq  1 + 1 IS  = Szeq (1 + η) ρS γ S ρS  

has been introduced.

(3.28)

Relaxation through Dipolar Interactions

53

Because of simultaneous action of double-resonance irradiation and dipole–dipole interaction, the steady-state solution for 〈Sˆ z 〉 is modified. The steady state 〈Sˆ z 〉 turns, after a 90° pulse, into a detectable signal, the intensity of which is multiplied by 1 + η. The phenomenon is referred to as the nuclear Overhauser enhancement (NOE). Before further discussion of the NOE (Section 3.3) and dipolar relaxation (Section 3.4), we wish to point out the formal similarity between the crossrelaxation (the magnetization exchange) and the chemical exchange. The Solomon equations expressed in the form of Equation (3.21) are formally very similar to McConnell’s modification of Bloch equation in the presence of two-site chemical exchange. We will come back to the Bloch–McConnell equations in Chapter 13. Moreover, we will discuss in more detail in Chapter 9 and Chapter 13 how the cross-relaxation and the chemical exchange can be observed in experiments of the same type.

3.3

The Nuclear Overhauser Enhancement

The nuclear Overhauser effect is one of the most interesting phenomena in NMR because it can be related to structural and dynamical properties of a molecule; studies of NOE are very important and common. We can discern two different situations: • The spins I and S are different species, e.g., 1H and 13C. In this case, we talk about heteronuclear NOE. This relaxation parameter carries information on relaxation mechanisms and on molecular dynamics. We will return to this point later. • The spins I and S are protons (homonuclear NOE). Although measurements of proton relaxation rates, ρI or ρS, are not very common because their interpretation is difficult, the proton NOE is today a very important parameter in NMR. The reason is that measurements of the homonuclear NOEs allow determination of molecular structure through the relation σIS ~ bIS2 ~ rIS–6 (see the following paragraphs). From the identification provided in Equation (3.22c), we see that, under certain conditions leading to W2 = W0, the cross-relaxation rate vanishes. Depending on the dynamics of the reorientation, the cross-relaxation rate and the homonuclear NOE can therefore be positive or negative. The homonuclear NOE effect can be explained in the following way. We can redraw the energy-level diagram of Figure 3.5 to yield Figure 3.6, in which the effect of saturation and relaxation has been indicated (adapted after Hore7). The saturating r.f. field at the resonance frequency of the I-spin tends to equalize population across the I-spin transitions, i.e., from the αα to the βα state and from the αβ to the ββ state (Figure 3.6b).

54

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

a

βI βS

βI αS

b

αI βS

βI αS

αI αS

c

W2

αI βS

αI αS

βI βS

βI αS

βI βS

d

αI βS

βI βS

βI αS

αI βS W0

α I αS

αI αS

FIGURE 3.6 Schematic diagram of the NOE effect. Filled circles indicate a population excess and open circles indicate a population deficit. (a) The situation at thermal equilibrium; (b) after r.f. irradiation on the I-spins, leading to saturation; (c) the effect of W2 relaxation; (d) the effect of W0 relaxation. Differences in level populations are proportional to signal intensities. (Adapted from Hore, P.J. Nuclear Magnetic Resonance, Oxford: Oxford University Press, 1995.)

Looking at the population differences across the S-spin transitions, the saturation of the I-spin does not yield any change by itself. Variation of the steady-state population difference and, consequently, of the signal intensity arises if the transition probabilities W2 and W0 are allowed to act in an unbalanced way. When the W2 dominates and the population transfer from ββ to αα is efficient, the net result of the saturation and relaxation is to increase the population differences across the S-spin transitions, a situation corresponding to Figure 3.6c. If, on the other hand, the W0 processes dominate, the population difference across the S-spin transitions is reduced and the signal intensity follows Figure 3.6d. The DD transition probabilities can be evaluated in terms of dipolar coupling constants and spectral densities corresponding to the angular variables in Equation (3.11a) through Equation (3.11f): W0 =

π 2 bIS J (ω I − ω S ) 5

(3.29a)

W2 =

6π 2 bIS J (ω I + ω S ) 5

(3.29b)

Transition probability (s−1)

Relaxation through Dipolar Interactions

55

10.0000 W0

1.0000 0.1000 0.0100

W2

0.0010 0.0001 0.0000 10−13

10−11

10−9

10−7

10−9

10−7

τc(s) (a) 1.0 0.5 η

0.0 −0.5

10−13

10−11

−1.0 −1.5

τc(s) (b)

FIGURE 3.7 The dependence of cross-relaxation rate on the correlation time. In (a) the transition probabilities W2 and W0 are shown. In (b) the η parameter as a function of the correlation time is displayed. A dipolar interaction strength bIS/2π = 15 kHz was used, based on an interproton distance of 2 Å.

The different numerical prefactors in the two expressions have their origin in the appropriate matrix elements of the spin operators occurring in the dipolar alphabet and in the normalization of spherical harmonics. If we assume that the spectral densities in Equation (3.29a) and Equation (3.29b) are given by Equation (3.16), then the relation between W2 and W0 at a given magnetic field depends only on the rotational correlation time, τc. From inspection of Equation (3.16), Equations (3.22), and Equations (3.29), we can thus see that the NOE depends on the correlation time of the molecules. This is shown in the diagrams of Figure 3.7. For small molecules with short correlation times, W0 < W2 (because of the larger numerical prefactor in W2) and the NOE is positive. For large molecules with long correlation times, the spectral density is frequency dependent and has a much higher value at the close-tozero frequency (ωS – ωI) than at the high frequency (ωS + ωI); consequently, W0 > W2 and a negative NOE is obtained. The diagrams are based on the specific and very simple motional model discussed so far — isotropic reorientation of the dipole–dipole axis in small, diffusional steps. We will discuss more sophisticated dynamic models in Chapter 6. However, the gross feature of Figure 3.7 can be believed to survive under more complicated motional conditions also.

56

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

For the sake of completeness, we also provide here the other two transition probabilities, obtained in a similar way:

3.4

W1I =

3π 2 bIS J (ω I ) 10

(3.30a)

W1S =

3π 2 bIS J (ω S ) 10

(3.30b)

Carbon-13 Relaxation

As mentioned earlier, the information content of proton T1 data is rather limited. This is mainly because many neighboring protons contribute to the relaxation process, the relaxation is not single exponential (unless under special conditions), and the resulting relaxation rates are not easily related to specific dynamic or structural parameters. The situation is different for carbon-13 and nitrogen-15. The natural abundance of these nuclides is low (1.1% for 13C, 0.37% for 15N) and, under the conditions of proton broadband irradiation, every carbon-13 or nitrogen-15 gives rise to a single line. For carbon-13 nuclei directly bonded to one or more protons, the dipole–dipole interaction with these protons is normally the dominant relaxation mechanism; other mechanisms can be significant for nitrogen-15. Application of proton broadband irradiation (spin decoupling) removes not only the J-couplings, but also the coupling between relaxation equations (compare Equations (3.19) or Equations (3.20)). Thus, the longitudinal carbon-13 magnetization is characterized by simple exponential relaxation (well-defined T1) and by an NOE factor, which is always positive. From Equation (3.26b) and Equation (3.28), we obtain: T1−C1 = ρC =

π 2  bCH  J (ω H − ω C ) + 3 J (ω C ) + 6 J (ω H + ω C )  5

γ  6 J (ω H + ω C ) − J (ω H − ω C ) η= H  γ C  J (ω H − ω C ) + 3 J (ω C ) + 6 J (ω H + ω C )

(3.31a)

(3.31b)

We use here the same normalization of spectral densities as in Equation (3.16) and the spectral densities as given in that equation can be inserted into Equation (3.31a) and Equation (3.31b). More sophisticated models based on more complicated molecular motion (discussed in Chapter 6) can also be applied. Measurements of carbon-13 relaxation parameters outside the extreme narrowing range, i.e., where they depend on the applied magnetic field, allow testing dynamic models and are in general an excellent source of information on molecular dynamics in solution. We will return to the application of the experiments of this type in Chapter 11.

Relaxation through Dipolar Interactions

57

Under extreme narrowing conditions, which here means that the product of the correlation time and the angular frequency must be much smaller than unity for all relevant frequencies (note that the highest frequency occurring in Equation 3.31a and Equation 3.31b is ωH + ωC), for a carbon-13 spin with a single directly bonded proton and assuming pure dipole–dipole relaxation, the situation becomes very simple indeed and the relaxation parameters are given by: 2 T1−C1 = bCH τc

η=

(3.32a)

γH = 1.99 2γ C

(3.32b)

For nitrogen-15, the situation is a little more complicated because it often has another relaxation mechanism (relaxation by chemical shift anisotropy; see Chapter 5) competing with the dipole–dipole relaxation. In addition, the magnetogyric ratio for nitrogen-15 is negative, which under certain conditions can lead to a signal intensity reduction by NOE. Due to the low sensitivity of the nitrogen-15, natural-abundance measurements are difficult and rarely performed. Nitrogen-15 relaxation measurements are, however, extremely important and useful for investigations of peptides and proteins; however, in this case, measurements are almost always carried out on isotopically enriched samples. Even though this section deals with heteronuclear dipolar relaxation, we should note that Equation (3.31a) and Equation (3.31b) are also valid for two unlike homonuclear spins. If we perform a selective I-decoupling experiment on a system of two protons, I and S, the longitudinal relaxation of the S-spin will follow an expression analogous to Equation (3.31a) and Equation (3.31b); the carbon and proton frequencies are replaced by the S and I proton Larmor frequencies, respectively. It is instructive to compare the spin-lattice relaxation rate in Equation (3.31a) and the corresponding homonuclear relaxation rate for the case of identical spins, which is obtained by setting the transition probability expressions of Equations (3.29) and Equations (3.30) into Equation (3.25). This yields: T1−H1 (ident) = 2

( )

(

)

3π 2  bHH J ω H + 4 J 2ω H    10

(3.33)

In the extreme narrowing limit, this takes the form: 2 T1−H1 (ident) = 23 bHH τc

(3.34)

Comparing the results for identical spins, Equation (3.34), to the previous situation with unlike spins, Equation (3.32a), we notice that the numerical prefactor is 3/2 times larger in the case of identical spins. This is the socalled “3/2 effect,” which implies that, everything else being equal, the relaxation efficiency of identical spins is larger than that for nonidentical

58

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

spins. The effect occurs because of the cross-relaxation between the identical spins — no longer directly observable, but still physically present. So far, we have only discussed the spin-lattice relaxation. Solomon also derived expressions for the spin-spin, or transverse, relaxation in the IS-spin system with dipole–dipole coupling. The derivation is not fully as elegant as for the longitudinal relaxation because one must work with the eigenstates of Iˆx instead of Iˆz . We will limit our interest here to the heteronuclear case and skip the details of the derivation (the derivation can be found in the original work1 or in the book by Hennel and Klinowski; see further reading). Also in this case, the relaxation of the I- and S-spins is coupled (biexponential). Under appropriate experimental conditions, discussed in Chapter 8, the transverse relaxation for the case of a carbon-13 relaxed by a directly bonded proton can become single exponential with the rate constant: T2−C1 =

π 2 bCH [ 4 J (0) + J (ω H − ω C ) + 3 J (ω C ) + 6 J (ω H ) + 6 J (ω H + ω C )] 10

(3.35)

We can see that the transverse relaxation rate contains the spectral density function taken at other frequencies than those present in the expressions for T1 and NOE, Equation (3.31a) and Equation (3.31b). It is therefore common that carbon-13 or nitrogen-15 relaxation studies also include the T2 measurements. Under extreme narrowing conditions (we note that here, also, the highest relevant frequency for which the condition ω 2τ 2c << 1 should be fulfilled is ωH + ωC), we obtain T1 = T2. The correlation-time dependence of carbon-13 relaxation rates at two magnetic fields, based on the simple isotropic and rigid model, is shown in Figure 3.8.

100.000

Relaxation rate (s−1)

1/T2 10.000 1.000 0.100

1/T1

0.010 0.001 10−13

10−11

10−9 τc(s)

10−7

FIGURE 3.8 1/T1 and 1/T2 relaxation rates calculated for a 1H-13C spin-pair using a rigid isotropic reorientation model for the spectral density function. Relaxation rates at 9.4 T are shown as solid lines, and relaxation rates at 18.8 T are shown as dashed lines. A value for the dipole–dipole coupling constant of 144·103 rad s–1 was used, which was obtained assuming an internuclear distance of 1.09 Å.

Relaxation through Dipolar Interactions

59

Equation (3.35) describes the correlation-time dependence of the natural linewidth for the 13C NMR signal, assuming an ideal, homogeneous magnetic field and neglecting spin couplings. As will be discussed later, the experimental determination of the spin–spin relaxation rate is usually not done by linewidth measurement, but rather by using multipulse techniques. The scalar carbon–proton couplings cannot in these experiments be suppressed in the same simple way as in the measurement of longitudinal relaxation and therefore special techniques are required. The relaxation rate corresponding to Equation (3.35) is then obtained in certain limiting situations and provided that certain precautions have been taken. We will return to experimental techniques for measuring longitudinal and transverse relaxation in Chapter 8.

3.5

Intermolecular Dipolar Relaxation

The dipolar interaction discussed so far in this chapter has been of intramolecular nature. Furthermore, the relaxation was assumed to be dominated by the dipole–dipole interaction with a directly bonded proton spin. This led, in a natural way, to the assumption of a constant internuclear distance and to the concept of the dipole–dipole coupling constant in the derivation of the time-correlation functions for the elements of the dipole–dipole Hamiltonian. The consequences of the fact that molecules are not rigid objects will be discussed in Chapter 6. We wish, however, to introduce here the concept of intermolecular dipolar interactions and intermolecular dipolar relaxation, which implies that the relevant internuclear distances are not constant. Consider a situation when a spin 1/2 nucleus has no close dipolar neighbor in the same molecule. This would be the situation, for example, for protons in chloroform (CHCl3) with the low natural abundance of carbon-13. In liquid chloroform, obviously a fairly high density of protons can interact with each other through the intermolecular dipolar interaction. The axis connecting protons in nearby molecules changes its orientation with respect to the magnetic field with time, in a similar way as in the intramolecular case. However, in addition, the relevant internuclear proton–proton distance or the strength of the dipole–dipole interaction changes randomly in time through translational diffusion. Another and more important example of intermolecular relaxation is encountered in solutions containing paramagnetic substances characterized by the presence of electron spin S (e.g., nitroxide radical or transition metal complexes), if they interact only weakly with the molecular species (solvent or solute) carrying the nuclear spins. The relevant translational motion in that case is the mutual diffusion of the molecular species containing the electron spins, S, and the nuclear spins, I. When dealing with a dipolar interaction modulated by translational diffusion, the time-correlation functions that we need to evaluate become more

60

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

complicated than the one encountered in Equation (3.15). Instead, we must work with:

Ginter (τ ) =

Y2 ,0 [Ω LIS (0)] Y2 ,0 [Ω LIS (τ )] rIS3 (τ ) rIS3 (0)

(3.36)

The symbol ΩLIS(τ) stands for the angles θ and φ specifying the orientation of the IS axis, at time τ, in the laboratory (L) coordinate frame. The symbol rIS(τ) denotes the distance between the spins, the length of the interspin vector rIS(τ). The general approach to calculating the Ginter(τ) is based on Equation (2.20) and requires formulating the conditional probability density for the vector rIS(τ) attaining a certain length and direction at a certain point in time, provided it took on given values at some other time. We will go through an example of such calculation in Chapter 6 for the rotational motion. For the translational case, we limit ourselves here to a sketch of a derivation; the reader is referred to the books of Abragam or McConnell (see further reading) for more detailed presentations. Briefly, the stationary conditional probability density, p(r0,0 | r,τ), of the interspin vector being r at time τ, provided that it was r0 at time zero, is obtained starting from the Fick’s law for translational diffusion in three dimensions. The relevant diffusion process is actually a mutual diffusion of the two spin-carrying molecules. The coefficient of mutual diffusion, D12, is equal to the sum of the diffusion coefficients for the two species; for the case of identical molecules, this is twice the “ordinary” translational diffusion coefficient. Abragam gives the conditional probability density as proportional to a Gaussian function:  (r − r0 )2  exp  −   4D12τ  As discussed independently by Hwang and Freed8 and by Ayant and coworkers9 in 1975, this expression is only valid for large r – r0. The correction for the proper behavior at small r – r0 (short diffusion time) is carried over in these works to the next step, calculation of the time-correlation function of Equation (3.36). After some rather tedious manipulations for the case of identical spins (i.e., a system similar to chloroform in our earlier example), the following expression for the time-correlation function was obtained for the simple case of molecules modeled as noninteracting, rigid, spherical particles with radii aI and with the spins in the centers of the spheres:

()

Ginter τ =

N I 18 d3 π

∞

∫ 0

 D 2  u2 exp  − 12 u τ  du 4 6 81 + 9u − 2 u + u  d2  2

(3.37)

Relaxation through Dipolar Interactions

61

where NI is the number density of spins I and d is the distance of closest approach between the spins, d = 2aI. It is useful to introduce a translational diffusion correlation time, τD, defined by:

τD =

d2 D12

(3.38)

A convenient expression for the spectral density can be derived10 corresponding to the time-correlation function of Equation (3.37), here in a dimensionless form: J inter (ω ) =

1 + 5 z/8 + z 2/8 1 + z + z 2/2 + z 3/6 + 4 z 4/81 + z 5/81 + z6/648

(3.39)

where z = (2ωτD)1/2. One should notice that the shape of the spectral density in Equation (3.39) is distinctly non-Lorentzian. The shape of the intermolecular spectral density is compared to the Lorentzian-like shape with the same correlation time, 1/(1 + ω2τ2D) in Figure 3.9. We can see in the figure that the shape of Jinter(ω) is characterized by a slower decay at high frequency. The intermolecular spectral density given by Equation (3.39) can also be used for the case of nuclear spins, I, being relaxed by electron spins, S. The electron spin relaxation is assumed to be much faster than the nuclear spin relaxation (which allows us to neglect the cross-relaxation effects), and yet much slower than the diffusion. The spin-carrying molecules are characterized by radii aI and aS and the diffusion coefficients DI and DS. The distance of closest approach becomes d = aI + aS and the mutual diffusion coefficient

5 × 10−9 Jintra

J(ω)

4 × 10−9 Jinter

3 × 10−9 2 × 10−9 1 × 10−9 100

103

106 ω(rad s−1)

109

1012

FIGURE 3.9 Comparison of the intermolecular spectral density function (Equation 3.39) and the Lorentzian spectral density given by Equation (3.16). A correlation time of 2 ns was used in both cases.

62

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

D12 = DI + DS. The intermolecular relaxation of I-spins can be shown to follow Bloch equations with the relaxation rates: 1 T1inter I

2

=

32π  µ0  γ 2γ 2  2S(S + 1) 1000 N A [ MS ] I S 7 J inter (ω S ) + 3 J inter (ω I )   405  4π  dD12

1 T2inter I

{

}

(3.40a)

2

=

16π  µ0  γ 2γ 2  2S(S + 1) 1000 N A [ MS ] I S   405  4π  dD12

{4 J

inter

}

(3.40b)

(0) + 13 J inter (ω S ) + 3 J inter (ω I )

NA is the Avogardro number and [MS] the molar concentration of the species carrying unpaired electrons. 1000NA[MS] replaces here the number density of spins (in m–3) in Equation (3.37). Several possible factors complicate the simple description of intermolecular dipolar relaxation by Equation (3.40a) and Equation (3.40b). First, the spins may reside off center in their molecules and the coupling between molecular rotational and translational motion needs then to be considered. This issue, referred to sometimes as spin eccentricity, has been studied by Ayant and coworkers.11 Second, the assumption of diffusing noninteracting hard spheres (called force-free diffusion) may not be a good approximation of dynamics in a real liquid. A more realistic description of a liquid is based on the statistical-mechanical concepts of distribution functions, in the simplest case of the spherical symmetry, a radial distribution function g(r) related to the potential of mean force, w(r) through: w(r ) = − kBT ln g(r )

(3.41)

The origin of this relation can be found in any book on statistical mechanics of liquids. Equation (3.37) and Equation (3.39) are based on the g(r) having the shape of a step function or the potential of mean force changing between zero and infinity at the distance of closest approach, d. The concept of radial distribution function was first introduced to the field of intermolecular relaxation by Harmon and Muller12 and explored further by Hwang and Freed.8 Hwang and Freed replaced Fick’s equation as a tool to obtain the conditional probability density by the Smoluchowski equation, which describes diffusion in a potential, such as that given in Equation (3.41). They proposed also an algorithm for implementing the effects of nonuniform distribution function on the intermolecular relaxation. The last complication to the simple description provided by Equation (3.40a) and Equation (3.40b) that we wish to mention at this point is relevant for the case of paramagnetic solutions. As mentioned earlier, the electron

Relaxation through Dipolar Interactions

63

spin relaxation is often very rapid. In some cases, it may be fast enough to compete with translational diffusion as the source of modulation of the intermolecular dipolar interaction. We will return to the difficult question of electron spin relaxation and its effects on nuclear spin relaxation in Chapter 15.

References 1. Solomon, I. Relaxation processes in a system of two spins, Phys. Rev., 99, 559–565, 1955. 2. Pake, G.E. Nuclear resonance absorption in hydrated crystals: fine structure of the proton line, J. Chem. Phys., 16, 327–336, 1948. 3. Spiess, H.W. Rotation of molecules and nuclear spin relaxation, NMR Basic Principles and Progress, 15, 59–214, 1978. 4. Smith, S.A., Palke, W.E., and Gerig, J.T. The Hamiltonians of NMR. Part I, Concepts Magn. Reson. 4, 107–144, 1992. 5. Smith, S.A., Palke, W.E., and Gerig, J.T. The Hamiltonians of NMR. Part II, Concepts Magn. Reson., 4, 181–204, 1992. 6. Smith, S.A., Palke, W.E., and Gerig, J.T. The Hamiltonians of NMR. Part III, Concepts Magn. Reson., 5, 151–177, 1993. 7. Hore, P.J. Nuclear Magnetic Resonance (Oxford: Oxford University Press), 1995. 8. Hwang, L-P. and Freed, J.H. Dynamic effects of pair correlation functions on spin relaxation by translational diffusion in liquids, J. Chem. Phys., 63, 40174025, 1975. 9. Ayant, Y., Belorizky, E., Alizon, J., and Gallice, J. Calculation of the spectral densities for relaxation resulting from random translational motion by magnetic dipolar coupling in liquids, J. Phys. (Paris), 36, 991–1004, 1975. 10. Polnaszek, C.F. and Bryant, R.G. Nitroxide radical induced solvent proton relaxation: measurement of localized translational diffusion, J. Chem. Phys., 81, 4038–4045, 1984. 11. Ayant, Y., Belorizky, E., Fries, P.H., and Rosset, J. Effect of intermolecular dipolar magnetic interactions on the nuclear relaxation of polyatomic molecules in liquids, J. Phys. (Paris), 38, 325–337, 1977. 12. Harmon, J.F. and Muller, B.H. Nuclear spin relaxation by translational diffusion in liquid ethane, Phys. Rev., 182, 400–410, 1969.

4 The Redfield Relaxation Theory

CONTENTS 4.1 The Eigenstate Formulation.......................................................................66 4.1.1 Equation of Motion for the Density Operator ...........................66 4.1.2 Introducing the Ensemble of Ensembles.....................................70 4.1.3 Approximations Leading to the Redfield Equation ..................72 4.1.4 The Physics of the Redfield Equation..........................................75 4.2 The Operator Formulation .........................................................................77 4.2.1 Irreducible Tensor Formulation of the Perturbation Hamiltonian .....................................................................................77 4.2.2 Equation of Motion of Expectation Values .................................84 4.2.3 Transverse Relaxation.....................................................................87 4.3 Relaxation in the Presence of Radiofrequency Fields ...........................89 References .............................................................................................. 94

So far, the description of relaxation has been focused on simple T1 and T2 relaxation processes and the somewhat more complicated phenomena of cross-relaxation and the nuclear Overhauser effect. A common feature of all these effects is that they are not related to the multiplet structure of NMR spectra. A more general formulation of relaxation theory, suitable also for systems with scalar spin–spin couplings (J-couplings), is known as the Wangsness, Bloch, and Redfield (WBR) theory or the Redfield theory.1–4 The Redfield formulation of relaxation theory has the advantage of being applicable to any type of nonequilibrium system and any relaxation mechanism. In analogy with the Solomon theory, the Redfield theory is also based on the second-order perturbation theory, which in certain situations (unusual for nuclear spin systems in liquids) can be a limitation. Rather than dealing with the concepts of magnetizations or energy level populations as in the Solomon formulation, the Redfield theory is given in terms of density operator. This allows a more general description of the nonequilibrium states and their evolution under random perturbation. The Redfield relaxation theory is often denoted as semiclassical theory. This means that the spin system is treated quantum mechanically, but the surroundings of the spins — sometimes called the lattice — with its stochastic processes 65

66

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

are dealt with classically. Two formulations of the Redfield theory are commonly used and we will go through both of them in the two following sections. In addition, the Redfield theory can also be expressed in the Liouville space formulation. This formulation is beyond the scope of this book and we refer to a review by Mayne and Scott in the NMR Encyclopedia.5 We finish this chapter by investigating the effects of radiofrequency fields on the nuclear spin relaxation.

4.1

The Eigenstate Formulation

In this section, we begin by formulating the equation of motion for the density operator for a system where the Hamiltonian can be written as a sum of a large time-independent part and a smaller time-dependent perturbation.

4.1.1

Equation of Motion for the Density Operator

In the semiclassical Redfield theory, the Hamiltonian is given as a sum of the main, time-independent part, Hˆ 0 , and a time-dependent perturbation, Hˆ 1 (t). The equation of motion of the density operator (the Liouville–von Neumann equation; compare Equation 1.16) is expressed by: dρˆ (t) = − i  Hˆ 0 + Hˆ 1 (t), ρˆ (t)  = i  ρˆ (t), Hˆ 0 + Hˆ 1 (t)  dt

(4.1)

The density operator is explicitly written as time dependent. If Hˆ 1 (t) = 0, the solution of Equation (4.1) can be written:

ρˆ (t) = exp(−iHˆ 0t)ρˆ (0)exp(+iHˆ 0t)

(4.2)

The exponential operators occurring in Equation (4.2) are defined by the series expansion:

exp(iFˆ ) = 1 + (iFˆ ) +

(iFˆ )2 (iFˆ )3 + + ... 2! 3!

(4.3)

in full analogy to the series expansion of the exponential function. We will return to a physical interpretation of the expression in Equation (4.2) a little later.

The Redfield Relaxation Theory

67

The influence of the time-independent part can be removed by transforming the Liouville–von Neumann equation to a new reference frame, the interaction frame. We will arrive at an expression for the time-dependent Hamiltonian in this frame. Let us define an operator ρˆ (t) so that:

ρˆ (t) = exp(+iHˆ 0t)ρˆ (t)exp(−iHˆ 0t)

(4.4a)

ρˆ (t) = exp(−iHˆ 0t)ρˆ (t)exp(+iHˆ 0t)

(4.4b)

or

By comparing Equation (4.4b) and Equation (4.2), we note that ρˆ (t) is constant and equal to ρˆ (0) if Hˆ 1 (t) = 0. Moreover, ρˆ (0) = ρˆ (0), independently of Hˆ 1 (t). For small Hˆ 1 (t), it is thus reasonable to expect that ρˆ (t) will change with time rather slowly; the transformation to the interaction representation removes the rapid variations of the density operator caused byHˆ 0 . Substituting Equation (4.4b) into the left-hand side of Equation (4.1) and comparing to the right-hand side, we obtain:

(

)

dρˆ (t) d  d = exp(−iHˆ 0t)ρˆ (t)exp(+iHˆ 0t)  = exp(−iHˆ 0t) ρˆ (t)exp(+iHˆ 0t) dt dt  dt

(

d d exp(+iHˆ 0t) + exp(−iHˆ 0t) ρˆ (t)exp(+iHˆ 0t) + exp(−iHˆ 0t)ρˆ (t) dt dt dρˆ (t) = −iHˆ 0 ρˆ (t) + exp(−iHˆ 0t) exp(+iHˆ 0t) + iρˆ (t)Hˆ 0 dt

) (4.5)

dρˆ (t) = i  ρˆ (t), Hˆ 0  + exp(−iHˆ 0t) exp(+iHˆ 0t) = i  ρˆ (t), Hˆ 0 + Hˆ 1 (t)  dt The commutator [ ρˆ (t), Hˆ 0 ] occurs now on both sides of the equation and can therefore be omitted. In this way we get: dρˆ (t) exp(−iHˆ 0t) exp(+iHˆ 0t) = i  ρˆ (t), Hˆ 1 (t)  dt

(4.6)

If we multiply Equation (4.6) by exp(+iHˆ 0t) from the left and by exp(−iHˆ 0t) from the right and use the fact that we can insert the product exp (−iHˆ 0t) exp(+iHˆ 0t) ≡ 1 between any pair of operators, we get a differential equation for dρˆ (t)/dt : dρˆ (t)  = i exp(+iHˆ 0t)ρˆ (t)exp(−iHˆ 0t), exp(+iHˆ 0t)Hˆ 1 (t)exp(−iHˆ 0t)  dt    = i  ρˆ (t), Hˆ 1 (t) 

(4.7)

68

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

where the following definition is introduced:  Hˆ 1 (t) = exp(+iHˆ 0t)Hˆ 1 (t)exp(−iHˆ 0t)

(4.8)

Equation (4.8) describes the transformation of Hˆ 1 (t) to the interaction representation. In the same way, ρˆ (t) is the interaction representation counterpart of the density operator, ρˆ (t) . The tilde above an operator (or its matrix element) denotes the interaction frame (or the interaction representation). It can be shown that the matrix elements of the operators follow the relations:

ρ αα ′ (t) = α ρˆ (t) α ′ = exp i(Eα − Eα ′ )t  α ρˆ (t) α ′

(4.9a)

and

( H (t))αα 1

′

 = α Hˆ 1 (t) α ′ = exp i(Eα − Eα ′ )t  α Hˆ 1 (t) α ′

(4.9b)

where |α〉 and Eα are an eigenfunction and an eigenvalue, respectively, of the main Hamiltonian, Hˆ 0 . By transforming the operators ρˆ (t) and Hˆ 1 (t) to the interaction representation, we remove the rapid oscillations in the off-diagonal matrix elements caused by Hˆ 0 , the main Hamiltonian. The diagonal elements in the two representations are identical. The transformation to the interaction representation has the same function as the transformation to the rotating frame, mentioned in the context of Bloch equations in Section 1.3. It separates the relatively uninteresting motion under Hˆ 0 from the effects of the perturbation. Assume that we deal with a system of isolated spins and that the unperturbed Hamiltonian is given by Hˆ 0 = −γ I B0 Iˆz (Equation 1.7). The transformation in Equation (4.8) then becomes: exp(−iγ I B0tIˆz )Hˆ 1 (t)exp(iγ I B0tIˆz ) = exp(+iω I tIˆz )Hˆ 1 (t)exp(−iω I tIˆz )

(4.10)

We note that the expression in Equation (4.10), generally written as: ˆ ′ = exp(−iθ Iˆ )A ˆ exp(iθ Iˆ ) A α α

(4.11)

(where  is an arbitrary operator), can be interpreted as a counterpart of the operator Â, rotated by the angle θ around the α-axis (see Levitt or Ernst et al. in the further reading section in the Preface). Thus, Equation (4.10)

The Redfield Relaxation Theory

69

describes the rotation of Hˆ 1 (t) around the z-axis by the angle γIB0t = −ωIt, i.e., the rotation around the field direction with the angular velocity |ωI| = γIB0. The differential equation for dρˆ (t)/dt , Equation (4.7), can formally be integrated from zero to time t: t

   ρˆ (t) = ρˆ (0) + i  ρˆ (t′), Hˆ 1 (t′)  dt′

∫

(4.12)

0

This is only a solution in the formal sense, because the ρˆ (t′) under the integral sign is not known. We can, however, try to obtain an approximate solution by setting ρˆ (t′) = ρˆ (0) . The validity of this may again be rationalized by noting that, for small Hˆ 1 (t), the variation of ρˆ (t) with time is slow. This yields: t

   ρˆ (t) = ρˆ (0) + i  ρˆ (0), Hˆ 1 (t′)  dt′   0

∫

(4.13)

In fact, it turns out (see following) that the term under the integral sign in Equation (4.13) vanishes and this equation is not a useful result as an approximate solution. A better approximation can, however, be obtained if we set ρˆ (t) given by Equation (4.13) into the formal solution, Equation (4.12): t  t′ ρˆ (t) = ρˆ (0) + i  ρˆ (0) + i  0  0 

∫

∫

    ˆ ˆ 1 (t ′′)  dt ′′ , H ˆ 1 (t ′)  dt ′ ( 0 ), ρ H      

t

 ˆ 1 (t ′)  dt ′ + (i)2 = ρˆ (0) + i  ρˆ (0), H  

∫ 0

t t′





 

(4.14) 

∫ ∫   ρˆ (0), Hˆ (t′′) , Hˆ (t′) dt′′dt′ 1

1

0 0

At this point, it is useful to find, once again, an expression for dρˆ (t)/dt:   dρˆ (t)  ˆ      = i  ρ(0), Hˆ 1 (t)  + (i)2   ρˆ (0), Hˆ 1 (t′)  , Hˆ 1 (t)  dt′ dt    t

∫

(4.15)

0

After making a variable substitution τ = t′ – t, Equation (4.15) becomes:   dρˆ (t)  ˆ      = i  ρ(0), Hˆ 1 (t)  −   ρˆ (0), Hˆ 1 (t + τ )  , Hˆ 1 (t)  dτ dt    t

∫ 0

(4.16)

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Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

Equation (4.16) describes in principle the time evolution of the density operator, which can be converted into an expression for derivatives of its matrix elements, d 〈α|ρˆ (t)|α ′〉/dt = dρ αα ′ (t)/dt (compare Equation 4.9a and Equation 4.9b). 4.1.2

Introducing the Ensemble of Ensembles

We begin by inspecting the contribution from the first commutator in Equation (4.16):    α  ρˆ (0), Hˆ 1 (t)  α ′ =  

∑ α ρˆ (0) β

 β Hˆ 1 (t) α ′

β

−

∑

 α Hˆ 1 (t) β

(4.17)

β ρˆ (0) α ′

β

Here, we employed the fact that, for a complete set of functions β, ∑ β|β 〉〈β|= 1 (the closure relation). Furthermore, the sum over the set of ket-bra products can be inserted between any pair of operators. Let us now consider an ensemble of systems (really, ensemble of ensembles; because we work with the density operator, we implicitly assume that our system is indeed an ensemble) identical at t = 0 (we assume that the perturbation is absent prior to t = 0), but characterized by different Hˆ 1 (t). Assume that the average of Hˆ 1 (t) over the ensemble of ensembles is zero at every point in time. If this requirement is not fulfilled for a particular choice of the perturbation Hamiltonian, then we can always modify the Hˆ 1 (t) by moving the average to the Hˆ 0 . As an example, consider the case of the dipole–dipole interaction acting as a perturbation. In an isotropic liquid, the average vanishes. In a weakly anisotropic liquid, the dipolar interaction gives rise to nonzero average of the Â-term in the dipolar alphabet, known as residual dipolar couplings. This averaged part of the interaction is not important from the point of view of relaxation and can be moved to Hˆ 0 (where it modifies the splitting term, in which the residual dipolar couplings are added to J-couplings). Consequently, we can write:  α Hˆ 1 (t) β = 0

(4.18)

In this particular case, we find it more convenient to use the horizontal bar to denote the ensemble average, because the symbol 〈…〉 denotes a matrix element. Because of the result of Equation (4.18), the first commutator in Equation (4.15) or Equation (4.16) does not contribute to dρˆ αα ′ (t)/dt. We can consider Equation (4.15) or Equation (4.16) as the first two terms in a series expansion in Hˆ 1 (t). In fact, by a similar argument, one can prove that all the odd terms in the expansion vanish and only even terms remain.

The Redfield Relaxation Theory

71

We proceed now by expanding the double commutator in Equation (4.16):

(

)

  ˆ      ˆ  ˆ ˆ ˆ ˆ   ρ(0), H1 (t + τ )  , H1 (t)  =  ρˆ (0)H1 (t + τ ) − H1 (t + τ )ρˆ (0) , H1 (t)            = ρˆ (0)Hˆ 1 (t + τ )Hˆ 1 (t) − Hˆ 1 (t + τ )ρˆ (0)Hˆ 1 (t)

(4.19)

    − Hˆ 1 (t)ρˆ (0)Hˆ 1 (t + τ ) + Hˆ 1 (t)Hˆ 1 (t + τ )ρˆ (0) The corresponding expression for the derivative of the matrix elements dρˆ αα ′ (t)/dt can be obtained by using Equation (4.18) and Equation (4.19) and the closure relation. For the ensemble of ensembles, we obtain: t

dρ αα ′ (t)   =− {〈α ρˆ (0) β 〉〈β Hˆ 1 (t + τ ) β ′ 〉〈β ′ Hˆ 1 (t) α ′〉 dt β ,β ′

∑∫ 0

  − 〈α Hˆ 1 (t + τ ) β 〉〈β ρˆ (0) β ′ 〉 〈β ′ Hˆ 1 (t) α ′〉 (4.20)   − 〈α Hˆ 1 (t) β 〉〈β ρˆ (0) β ′〉〈β ′ Hˆ 1 (t + τ ) α ′〉   + 〈α Hˆ 1 (t) β 〉〈β Hˆ 1 (t + τ ) β ′〉〈β ′ ρˆ (0) α ′〉}dt′ Now, let us see how we can use this result to compute transition probabilities. Before proceeding with the general case, let us first assume that we deal with a system with two possible states (k and m) that is initially in the state |k〉, i.e., we have an ensemble of pure states |k〉. This implies that the initial density matrix elements are: k ρˆ (0) k = k ρˆ (0) k = 1

(4.21a)

m ρˆ (0) n = 0 unless m = n = k

(4.21b)

and

Now, we want to compute the transition probability Wk→m. The transition probability is given by the rate of change of the matrix element 〈m|ρˆ (t)|m〉 (compare Equation 4.9a): Wk→m =

d dρ (t) m ρˆ (t) m = mm dt dt

(4.22)

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Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

Equation (4.20) can be used to evaluate Equation (4.22), given the initial conditions of Equations (4.21). These conditions have the effect of making the first and last terms of Equation (4.20) vanish. The remainder is: t

Wk→m =



∫ { m Hˆ (t + τ ) k 1

  k Hˆ 1 (t) m + m Hˆ 1 (t) k

 k Hˆ 1 (t + τ ) m }dτ

0 t

=

∫

m Hˆ 1 (t + τ ) k

k Hˆ 1 (t) m exp(iω mk τ )dτ

0

(4.23)

t

+

∫

m Hˆ 1 (t) k

k Hˆ 1 (t + τ ) m exp(−iω mk τ )dτ

0

t

=

∫V

mk

(t)Vkm (t + τ )exp(iω kmτ )dτ + c.c.

0

where the identification Em − Ek = ωmk and Vmk (t) = 〈m|Hˆ 1 (t)|k 〉 has been made. The averaging implied by the overbar is performed over different perturbations, Hˆ 1 (t) . The expression for the transition probability is identical to the result of the second-order time-dependent perturbation theory, * Equation (2.37). The operator Vˆ = Hˆ 1 (t) is hermitian; that is, Vmk = Vkm, and c.c. in Equation (4.23) means complex conjugate also here.

4.1.3

Approximations Leading to the Redfield Equation

We now return to Equation (4.16) for a description of a more general case. In order to proceed, we need to make an important assumption. We assume that motions in the environment, or surrounding, of the spins (the “lattice” or the “bath”) are characterized by a short correlation time, τc. The correlation time should be sufficiently short to make its product with the root-mean square fluctuation of the perturbation — here somewhat loosely defined as ω SB =|H12 |1/2 — much less than unity:

ω SBτ c << 1

(4.24)

The inequality in Equation (4.24) defines the conditions of motional narrowing (not to be confused with extreme narrowing). We concentrate on the behavior of the density operator on the time scale t, so that: −1 τ c << t << ω SB

(4.25)

Under these conditions, it is legitimate to stop our expansion of dρˆ (t)/dt after the second commutator, as in Equation (4.16). Each further term would

The Redfield Relaxation Theory

73

be related to the foregoing one roughly as ωSBτc, which, according to Equation (4.25), is much less than one. In order to come to useful expressions, we need some further simplifying assumptions: • We replace ρˆ (0) in the right-hand side of Equation (4.16) by ρˆ (t) . We can do this because of the assumed slow variation in ρˆ (t) and the shortness of the period t.  • We neglect the possible correlations between ρˆ (t) and Hˆ 1 (t) . This makes sense because of the separation of time scales, again resulting from Equation (4.25). • We extend the upper limit of the integration to infinity. We can do this because the short correlation time means that the integrand vanishes for long times anyway. Upon these assumptions, we obtain the following “master equation”:   dρˆ (t)    = −   ρˆ (t), Hˆ 1 (t + τ )  , Hˆ 1 (t)  dτ dt    0  ∞

∫

(4.26)

∞

   = −  Hˆ 1 (t),  Hˆ 1 (t + τ ), ρˆ (t)   dτ     0 

∫

Strictly speaking, we should also use an overbar over the density matrix element in the left-hand side of Equation (4.20) and Equation (4.26). This would complicate the notation in the rest of this chapter and we choose not to do it, assuming instead implicitly that the density operator and its matrix elements refer to the average density operator, defined in the ensemble of ensembles. Equation (4.26) can be rewritten in terms of matrix elements of ρˆ (t) , using the basis set of the eigenstates to Hˆ 0 . After somewhat tedious calculations, one obtains: dρ αα ′ (t) = dt

∑ exp i(α − α ′ − β + β′)t  R

αα ′ ββ ′  ββ ′ (t)

ρ

(4.27)

β ,β ′

where ρ ββ ′ (t) = 〈β|ρˆ (t)|β ′〉 , α ≡ Eα (in angular frequency units) and the exponential factor has its origin in Equation (4.9b). The symbols Rαα ′ββ ′ are called relaxation supermatrix elements and are given by: Rαα ′ββ ′ = 1 [Jαβα ′β ′ (α ′ − β ′) + Jαβα ′β ′ (α − β ) 2 − δα ′β ′

∑J γ

γ − β ) − δαβ

γ βγα (

∑J γ

γ − β ′)]

γα ′γ β ′ (

(4.28)

74

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

where we again explicitly average over the ensemble of different perturbation Hamiltonians: ∞

Jαα ′ ββ ′ (ω ) = 2

∫ α Hˆ (t) α ′ 1

β ′ Hˆ 1 (t + τ ) β exp(−iωτ )dτ

(4.29)

0

In fact, the functions Jαα,ββ(ω) as defined previously may be complex. The real part is normally much larger than the imaginary part. The imaginary part gives rise to small shifts in line frequencies, called dynamic frequency shifts. We will return to this issue in the next section. As a further approximation, we reduce the number of relaxation supermatrix elements by retaining only those for which α – α ′ – β + β ′ = 0, i.e., only those for which exp[i(α – α ′ – β + β ′ )t] = 1. These terms are called secular and the approximation is known as the secular approximation (note that the term “secular” is used here in a meaning different from the one in Section 3.1). The nonsecular terms are less efficient in causing relaxation because they oscillate rapidly, which effectively averages them down to (almost) zero. In particular, the secular approximation prevents coupling of density matrix elements corresponding to different orders of coherence, pαα ′ = Mα – Mα ′ (where Mα and Mα′ are the total magnetic quantum numbers for the states α and α ′). The issue of the order of coherence is important in modern NMR and we will return to it in Chapter 7. Within a block of a given coherence order, the secular approximation should not be used if one has nearly degenerate transitions, α − α ′ ≈ β − β ′, i.e., overlapping lines. Making use of the secular approximation, Equation (4.27) takes the form: dρ αα ′ = dt

∑R

αα ′ ββ ′

ρ ββ ′

(4.30a)

β ,β ′

where the summation is restricted to the secular terms. Equation (4.30a) may be rewritten in a form making explicit use of the relaxation superoperator: dρˆ (t) ˆˆ ˆ = Rρ(t) dt

(4.30b)

If required, one can formulate an equation similar to Equation (4.30a) for the density matrix elements in the laboratory frame rather than in the interaction representation. This can be done without invoking the secular approximation. We use the facts that:

ρ αα ′ = exp(i(α − α ′)t)ραα ′

(4.31)

The Redfield Relaxation Theory

75

and dρ αα ′ dρ = i(α − α ′)ρ αα ′ + exp(i(α − α ′)t) αα ′ dt dt =

∑ β ,β ′

Rαα ′ ββ ′ exp i(α − α ′)t  ρββ ′

(4.32)

In the second equality, we make use of Equation (4.27). The laboratory frame counterpart of Equation (4.27) becomes thus: dραα ′ = i(α ′ − α )ραα ′ + dt

∑R

αα ′ ββ ′

ρββ ′

(4.33)

β ,β ′

We note that the expressions for the time evolution of diagonal elements of the density matrix (populations), α = α ′, in the interaction representation and in the laboratory frame are identical. Differences arise in the expressions for time evolution of coherences. The laboratory frame formulation has an advantage of not invoking the secular approximation. This feature becomes particularly useful in the context of simulations of NMR lineshapes in systems in which the frequency splittings in the spectra are not very much larger than the relaxation supermatrix elements. Examples of such a treatment can be found in a paper by Werbelow and Kowalewski.6 We will come back to this point in Chapter 14.

4.1.4

The Physics of the Redfield Equation

Consider now the physical meaning of the various relaxation supermatrix elements. Rααββ are simply the transition probabilities between the levels α and β. The spin-lattice relaxation (T1 processes) is related to these elements. The symbols Rαβαβ describe the relaxation of the off-diagonal elements of the density operator, i.e., coherences between states α and β, or T2-type processes. As a consequence of the secular approximation, we obtain that the relaxation of nondegenerate (and nonoverlapping) lines is a simple exponential process and the lines are Lorentzian. Summarizing the properties of various relaxation supermatrix elements, we come to the conclusion that the matrix representation of the relaxation superoperator can be given a pictorial description shown in Figure 4.1 (based on the book by Ernst et al. (further reading); and known as the “Redfield kite”). Before leaving this section, we wish to discuss two rather subtle but important −1 questions. The first one is related to the condition t << ω SB that we assumed in order to obtain the master equation. The relaxation times in NMR (Rαα ′ββ ′–1) are −1 typically much longer than ω SB and we are normally interested in following the behavior of the density operator on this rather long time scale. This dilemma

76

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications 1QT

2QT

2QT

1QT

ZQT

Populations

Populations ZQT

FIGURE 4.1 The Redfield kite. The Redfield relaxation matrix has a block structure due to the neglect of nonsecular terms. Solid squares indicate nonzero relaxation rate constants between different magnetization orders. Only populations have nonzero cross-relaxation rate constants. The dashed boxes indicate additional nonzero cross-relaxation rates if transitions are degenerate.

can be resolved if we consider our long time period as a sequence of short −1 intervals τB, so that τ B << t << ω SB . Because Hˆ 1 (t) in the consecutive intervals can safely be assumed to be uncorrelated, we can generalize the validity of the master equation to longer t. The remaining validity condition for the Redfield theory is then Expression (4.24). If this condition is fulfilled, we always find: Rαα′ββ ′–1 >> τc

(4.34)

That is, the relaxation times are always longer than the correlation time. Some authors consider this relation to be the validity condition of the Redfield theory, which is not fully correct. The second point is related to the effect of nonequal equilibrium populations of different levels on transition probabilities, an issue mentioned in Section 3.5. Expression (4.28) leads to Rααββ = Rββαα or to the transition probabilities α → β and β → α being equal. This in turn leads to an equal distribution of populations between different levels, which corresponds to the equilibrium distribution at infinite temperature. As stated earlier, to get a correct description at finite temperatures, one should in principle treat the lattice quantum mechanically. In practical terms, this leads to a modification of Equation (4.30a): dρ αα ′ = dt

sec.

∑R

αα ′ ββ ′ (  ββ ′

β ,β ′

ρ

eq − ρ ββ ′ (T ))

(4.35)

The Redfield Relaxation Theory

77

eq where the summation is restricted to secular terms and ρ ββ ′ (T ) denotes the equilibrium density matrix element at temperature T, given by Equation (1.18) for the diagonal elements (populations) and equal to zero if β ≠ β′. Because the diagonal (β = β′ ) matrix elements are identical in the interaction representation and the laboratory frame, we can write:

eq ρββ ′ (T ) = δ ββ ′

exp(− β/kBT )

∑ exp(−β′′/k T)

(4.36)

B

β ′′

It is worth noting the similarity of Equation (4.35) to Equation (3.21) because it provides a link between the simple, Solomon-like approach and the more complete Redfield treatment.

4.2

The Operator Formulation

In practical calculations, we are usually not so much interested in the evolution of the density matrix as in that of observable quantities, represented by quantum mechanical operators. Therefore, a formulation of the Redfield theory in terms of evolution of observable quantities has certain advantages. An extensive presentation of the operator formulation of the Redfield relaxation theory has been given in Abragam’s and Goldman’s books (further reading) and in a review by Goldman.7 A shorter presentation and a comparison with the eigenstate formulation has been given by Murali and Krishnan.8 4.2.1

Irreducible Tensor Formulation of the Perturbation Hamiltonian

In analogy with the eigenstate formulation, the master equation, Equation (4.26), is the starting point for further discussion. To proceed, we introduce here a general expression for Hˆ 1 (t) , similar to Equation (3.13): l

Hˆ 1 (t) = ξ

∑ (−1) V q

l ,q

(t)Tˆl ,− q

(4.37a)

q=− l

The perturbation is expressed as a sum of 2l + 1 terms, where l is called the tensor rank (for the DD interaction described in Chapter 3, we have l = 2). Every term is a product of a classical random function of time, ξVl,q(t), and an expression, Tˆl ,− q , dependent only on the spin operators. The strength of the interaction, ξ, has here been factored out from the function of time. For the DD interaction between the spins I and S, we will use the notation ξISDD. Thus, Equation (4.37a) formulates clearly the semiclassical nature of the theory.

78

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

The functions Vl,q(t) are in general complex, fulfill V*l,q(t) = (–1)qVl,–q(t), and depend on the space variables, physical constants, and time. The operator Tˆl ,q is called the q-th component of the rank-l irreducible tensor operator and fulfills Tˆl†,q = (−1)q Tˆl ,− q , where Tˆl†,q is called hermitian conjugate of Tˆl ,q . We find it useful to express Hˆ 1 (t) also in terms of the complex and hermitian conjugates of the quantities in Equation (4.37a): l

Hˆ 1 (t) = ξ

∑ (−1) V q

* l ,q

(t)Tˆl†,− q

(4.37b)

q=− l

The reason for commonly choosing to work in terms of the irreducible tensor operators is that they have convenient behavior under rotation of the axes of the coordinate system. In fact, the irreducible tensor operators are defined according to their properties under such rotations: l

Tˆl′,q ( new axes) =

∑ Tˆ

l,p

(old axes)Dpl ,q (α , β , γ )

(4.38)

p=− l

The prime refers to the new set of axes. The symbols α, β, and γ are the three Euler angles necessary to define an arbitrary rotation of the coordinate frame. The Euler angles are defined in Figure 4.2. For further discussion of the

FIGURE 4.2 Illustration of the rotations in three dimensions by the three Euler angles, α, β, and γ. The original axes are denoted x, y, and z. The rotations are defined according to the convention that uses first a rotation about the z-axis by an angle α. This leads to the new positions/axes (x′,y′,z′). Next, one performs a rotation by β about the x′-axis (leading to the x″, y″, z″ positions) and, finally, a rotation by g around the new z″-axis yields the final positions x″′, y′″, z″′.

The Redfield Relaxation Theory

79

subject, consult the book by Brink and Satchler (see further reading). The symbol Dlp,q(α,β,γ ) denotes the elements of the Wigner rotation matrix. The matrix elements have a simple dependence on the angles α and γ : Dpl ,q (α , β , γ ) = exp ( −i(α p + γ q)) dpl ,q (β )

(4.39)

The β -dependence is more complicated. The dlp,q(β ) functions for l = 2 are listed in Table 4.1; for further details, see the book by Brink and Satchler. A second important property of irreducible tensor operators is a consequence of the relation between rotations of the coordinate systems and angular momentum operators, illustrated in Equation (4.11). As a consequence of the transformation properties under rotations, the irreducible spherical tensor operators follow the relation:  Iˆz , Tˆl ,q  = qTˆl ,q  

(4.40a)

It is easy to convince oneself (readers are advised to do so) that Relation (4.40a) is fulfilled by the operators: Iˆ10 = Iˆz Iˆ1±1 = ∓

1 2

(4.41a) Iˆ±

(4.41b)

where Iˆ± are defined in Equation (3.12) and we recall the relations:  Iˆz , Iˆ±  = ± Iˆ±  

(4.40b)

derived in Atkins’ and Friedman’s book (see further reading). We note that Iˆ+ and Iˆ− are hermitian conjugates of each other. One should add that the choice of the operator in Equation (4.41a) and Equation (4.41b) is not unique; we follow here (and in the following) the convention of Brink and Satchler. The third useful property of the irreducible tensor operators is the fact that the operators with ranks l1 and l2 can be coupled together to form operators of rank l = l1 + l2, l1 + l2 – 1,…, |l1 – l2|. Consider two irreducible tensor operators, with elements Rˆ l1 ,q1 and Pˆl2 ,q2 . The elements Tˆl ,q of the irreducible tensor operator of rank l (corresponding to one of the values given previously) are given by:

Tˆl ,q =

l1

l2

∑ ∑ Rˆ

Pˆ

l1 , q1 l2 , q2

q1 =− l1 q2 =− l2

C(l1l2 q1q2|lq)

(4.42)

80

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications TABLE 4.1 The dlp,q(β) Functions for l = 2 dlp,q(β)

p

q

–2

–2

cos 4 (β/2)

–2

–1

1 2

–2

0

–2

1

–2

2

–1

–2

–1

–1

–1

0

–1

1

–1

2

0

–2

3/8 sin 2 β

0

–1

− 3/2 sin β cos β

0

0

0

1

3/2 sin β cos β

0

2

3/8 sin 2 β

1

–2

1 2

sin β(cos β − 1)

1

–1

1 2

(2 cos β + 1)(1 − cos β )

1

0

1

1

1 2

(2 cos β − 1)(cos β + 1)

1

2

1 2

sin β(1 + cos β )

2

–2

sin 4 (β/2)

2

–1

1 2

2

0

2

1

2

2

sin β(1 + cos β ) 3/8 sin 2 β

− 21 sin β(cos β − 1) sin 4 (β/2) − 21 sin β(1 + cos β ) 1 2

(2 cos β − 1)(cos β + 1) 3/2 sin β cos β

1 2

(2 cos β + 1)(1 − cos β )

− 21 sin β(cos β − 1)

1 2

(3 cos 2 β − 1)

− 3/2 sin β cos β

sin β(cos β − 1) 3/8 sin 2 β

− 21 sin β(1 + cos β ) cos 4 (β/2)

The Redfield Relaxation Theory

81

where C(l1l2q1q2lq) is a vector coupling coefficient or a Clebsch–Gordan coefficient. Exactly the same coefficients occur in the theory of coupling of angular momenta when the subsystem eigenfunctions are combined to form eigenfunctions for the total angular momentum. See the book by Brink and Satchler (further reading) for further information and explicit expressions for the Clebsch–Gordan coefficients. Let us identify Rˆ l1 ,q1 with the operators given in Equation (4.41a) and Equation (4.41b) and Pˆl2 ,q2 with the analogous operators for the S-spin. The rank-2 irreducible tensor operators obtained using Equation (4.42) and the appropriate Clebsch–Gordan coefficients are as follows: Tˆ2 ,0 =

1 6

(3 Iˆz Sˆ z − Iˆ ⋅ Sˆ ) =

1 6

 2 Iˆz Sˆ z − 1 ( Iˆ+ Sˆ − + Iˆ− Sˆ + )  2  

(4.43a)

Tˆ2 ,±1 = ∓ 21 ( Iˆ±Sˆ z + IˆzSˆ ± )

(4.43b)

Tˆ2 ,± 2 = 21 Iˆ±Sˆ ±

(4.43c)

To construct the Hamiltonian, Hˆ 1 (t) , defined in Equation (3.13) and Equation (4.37a) with which we started this section, we use this form of the irreducible tensor operators together with the functions Vl,q equal to normalized spherical harmonics, fulfilling the definitions of irreducible tensors:  5  Y2 ,0 =    16π   15  Y2 ,±1 = ∓    8π 

1/2

( 3 cos2 θ − 1)

(4.44a)

cos θ sin θ exp(±iφ )

(4.44b)

1/2

 15  Y2 , ± 2 =   32π 

1/2

sin 2 θ exp(± 2iφ )

(4.44c)

In this way, we can obtain the relation between the dipolar interaction strength constant ξISDD and bIS, defined in Chapter 3. By multiplying together the normalization constants in Equations (4.43) and Equations (4.44) and comparing the prefactors of Equation (3.10) and Equations (3.11) to the DD version of Equation (4.37a), we obtain the relationship:

() ( )

bIS = ξISDD 1 6

1/2

2

5 16π

1/2

= ξISDD

( ) 5 24π

1/2

(4.45)

82

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

Let us now transform Hˆ 1 (t) , given in Equation (4.37a), to the interaction representation, according to Equation (4.8):  Hˆ 1 (t) = exp(iHˆ 0t)Hˆ 1 (t)exp(−iHˆ 0t) =ξ

∑(−1) V q

l ,q

(4.46)

(t)exp(iHˆ 0t)Tˆl ,− q exp(−iHˆ 0t)

q

The expression exp(iHˆ 0t)Tˆl ,− q exp(−iHˆ 0t) requires careful consideration. Let us consider Hˆ 1 (t) as the dipolar Hamiltonian of Equation (3.13), with the spin operators in the form of Equation (4.43a), Equation (4.43b), and Equation (4.43c) and Hˆ 0 consisting of the Zeeman interaction for the spins I and S, as given in Equation (3.7). We note that the operators Iˆz and Sˆ z commute with each other, which has the consequence that: exp(iHˆ 0t) = exp(iω I tIˆz )exp(iω StSˆ z )

(4.47)

The expression exp(iHˆ 0t)Tˆl ,− q exp(−iHˆ 0t) becomes thus: exp(iHˆ 0 t)Tˆl ,− q exp(−iHˆ 0 t)

(4.48)

= exp(iω I tIˆz )exp(iω StSˆ z )Tˆl ,− q exp(−iω I tIˆz )exp(−iω StSˆ z )

At this point, it is again useful to recall Expression (4.11). According to this expression, Equation (4.48) implies that the operators Tˆl ,− q are rotated around the z-axis. More specifically, the I-spin operators in Tˆl ,− q are rotated according ˆ ′ = exp(iω tIˆ )A ˆ exp(−iω tIˆ ) , while the S-spin operators in Tˆ are rotated to A I z I z l ,− q by the Zeeman Hamiltonian of the S-spin. The exponential operators containing Iˆz do not influence the S-spin operators (and vice versa) because S-spin operators and I-spin operators always commute. As a result, the exponential operators containing Îz introduce, according to Equation (4.48), terms rotating at the positive and negative I-spin Larmor frequency, as well as zero-frequency terms. In the same way, the exponential operators containing Sˆ z lead to the occurrence of positive and negative S-spin Larmor frequencies. For example, explicit calculations for q = 0 yield: exp(iHˆ 0t)Tˆ2 ,0 exp(−iHˆ 0t) =

1 6

(

)

(

)

 2 IˆzSˆ z − 1 Iˆ+Sˆ − exp i(ω I − ω S )t − 1 Iˆ−Sˆ + exp −i(ω I − ω S )t  2 2  

(4.49)

The expression exp(iHˆ 0t)Tˆl ,− q exp(−iHˆ 0t) for any q becomes: exp(iHˆ 0t)Tˆl ,− q exp(−iHˆ 0t) =

∑ Aˆ

( − q) p

p

exp(iω p( − q)t)

(4.50)

The Redfield Relaxation Theory

83

ˆ ( − q) are a set of operators and the frequencies ω ( − q) correspond to Here, A p p zero or to linear combinations of the I- and S-spin Larmor frequencies. The symbols on the right-hand-side of Equation (4.50) are summarized in Table 4.2. The expressions given in Equation (4.46) and Equation (4.50) can be inserted in the “master equation” (4.26). We make use of Equation (4.37a) for Hˆ 1 (t) and of Equation (4.37b) for Hˆ 1 (t + τ ) and note that, in an ensemble ˆ ( q) are identical, and the functions V (t) of systems, all the operators Tˆl ,− q or A p l,q are random. After ensemble averaging, this leads to: dρˆ (t) ˆ ( − q) ,  A ˆ †( − q′ ) , ρˆ (t)   (−1)q+ q′ exp i(ω p( − q) + ω p( −′ q′ ) )t  A =− p    p′  dt q , q′ , p , p′

(

∑

)

(4.51)

∞

× ξ2

∫V

l ,q

(t)Vl*,q′ (t + τ ) exp(iω p( −′ q′ )τ )dτ

0

We recognize the term under the integral sign (where the symbol 〈…〉 has the same meaning as the overbar in Equation 4.18, Equation 4.20, Equation 4.23, Equation 4.26, or Equation 4.29) as a time-correlation function. From the fact that Vl,q(t) are defined as irreducible spherical tensors, the corresponding time-correlation functions follow the symmetry relation9: Vl ,q (t)Vl*,q′ (t + τ ) = Gl ,q ,q′ (τ ) = δ q ,q′Gl (τ )

(4.52)

We will return to this relation in Chapter 6.

TABLE 4.2 Results of Calculations of Symbols in the Right-Hand-Side of Equation (4.50)

ω p( q)

Ap( q) q

p=1

–2

1 2

Iˆ− Sˆ −

–1

1 2

Iˆ− Sˆ z Iˆ Sˆ

0

2 3 z z

1

− 21 Iˆ+ Sˆ z

2

1 2

Iˆ+ Sˆ +

1 2

−

p=2

p=3

p=1

p=2

p=3

—

—

−ω I − ω S

—

—

IˆzSˆ −

—

−ω I

−ω S

0

ω I − ωS

—

ωI

ωS

—

ω I + ωS

Iˆ Sˆ

1 24 + −

− 21 IˆzSˆ + —

−

Iˆ Sˆ

1 24 − +

—

−ω I + ω S —

—

—

84

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

Using Relation (4.52) and neglecting nonsecular terms, Equation (4.51) simplifies to: dρˆ (t) = − 1 ξ2 dt 2

∑ j (−ω l

q,p

( − q) p

ˆ ( − q) ,  A ˆ †( − q) , ρˆ (t)   )  A p  p   

(4.53)

We call this formulation “master equation in operator form.” Note that we deal here with the spectral densities, jl (ω p( − q) ), related to the classical random functions, Vl,q(t). These are not the same as the spectral densities occurring in Equation (4.28) and Equation (4.29), which contain the matrix elements of the full perturbation Hamiltonian. The matrix elements of the spin operators, present in the spectral densities in those equations, occur in Equation (4.53) in the double commutator. In analogy with the spectral densities J(ω), the jl(ω) are indeed complex quantities, jl(ω) = Jl(ω) + iLl(ω). The real part is important for relaxation and usually much larger than the imaginary part, which leads to dynamic frequency shifts, small shifts in line positions. The real part of the spectral density is an even function of frequency, Jl(ω) = Jl(–ω); the imaginary part is an odd function of frequency, Ll(ω) = –Ll(–ω). In most of the discussions to follow, we will neglect the dynamic frequency shifts and use the symbol jl(ω), which we simply call “spectral density.” The reader interested in the dynamic frequency shifts is advised to consult the reviews by Werbelow and London.10,11 4.2.2

Equation of Motion of Expectation Values

The operator approach becomes really useful when we move from looking at the time derivative of the density operator to studying the time dependence of an expectation value of an operator corresponding to a measurable quantity. The relation between the time-dependent expectation value of an operator Qˆ , denoted q(t), and the density operator was formulated in Chapter 1 (Equation 1.17), stating that q(t) = 〈Qˆ 〉 (t) = Tr( ρˆ (t)Qˆ ) (in the Schrödinger representation). The interaction representation counterpart reads: q(t) = Tr( ρˆ (t)Qˆ ) = Tr(Qˆ ρˆ (t))

(4.54)

where we have applied one of many useful relations between traces of ˆ ˆ ) = Tr(BA ˆ ˆ ). products of operators: Tr( AB Note that Equation (1.17) is equally valid in the laboratory frame and in the interaction frame. Thus, in order to obtain the expectation value of Qˆ in the interaction frame, we express the density operator in that frame. Most of the time, the expectation value q(t) can be interpreted as an expectation

The Redfield Relaxation Theory

85

value in the rotating frame. Taking the time derivative of both sides of Equation (4.54), we obtain:   d d q(t) = Tr Qˆ ρ (t)  dt   dt

(4.55)

where we have used the fact that the operator Qˆ is not dependent on time. By substituting the master equation (4.53) into the right-hand side of Equation (4.55) and neglecting the imaginary part of the spectral density, the following expression is obtained:  d  q(t) = − 1 ξ 2 Tr  dt 2 

 ˆ ( − q) ,  A ˆ †( − q) , ρˆ (t)    Jl −ω p( − q) Qˆ  A p  p    

∑ ( q,p

)

(4.56)

We proceed by making use of another relation between traces:

{

} {

}

ˆ  Bˆ ,Cˆ  = Tr  A ˆ ˆ ˆ Tr A  , B  C  

(4.57)

Applying this twice to Equation (4.56), we obtain:  d  q(t) = − 1 ξ 2 Tr  dt 2    = − 1 ξ 2 Trr  2 

∑

 ˆ ( − q)   A ˆ †( − q) , ρˆ (t)   J l (−ω p( − q) ) Qˆ , A p p      

∑

 ˆ ( − q)  , A ˆ †( − q)  ρˆ (t)  J l (−ω p( − q) )  Qˆ , A p p    

q,p

q,p

(4.58)

= − Tr{ Pˆ ρˆ (t)} = − p (t) where the operator Pˆ (with a time-dependent expectation value in the rotating frame, p (t)) has been defined as: Pˆ = 1 ξ 2 2

∑ J(−ω q,p

( − q) p

ˆ ( − q)  , A ˆ †( − q)  )  Qˆ , A p  p  

(4.59)

Equation (4.58) states that the time derivative of an expectation value of an operator corresponding to an observable is equal to an expectation value of another operator. As stated in Equation (4.59), this other operator is a

86

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

ˆ ( − q) ], A ˆ †( − q) ] with the spectral linear combination of double commutators [[Qˆ , A p p ( − q) densities, Jl (−ω p ) , as the coefficients. Equation (4.58) has the same deficiency as Equation (4.27) and Equations (4.30): it requires the correction for the finite temperature (compare Section 2.5). We define the expectation value of the operator Pˆ at thermal equilibrium at finite temperature as:   peq = 〈 Pˆ 〉eq = Tr( Pˆ ρˆ eq ) = Tr  1 ξ 2  2

∑ q,p

 ˆ ( − q)  , A ˆ †( − q)  ρˆ eq  J l (−ω p( − q) )  Qˆ , A p p    

(4.60)

and modify Equation (4.58) to read: d q(t) = −( p (t) − peq ) dt

(4.61)

As an illustration of the formalism, let us go back to the specific case of the dipole–dipole interaction between unlike spins I and S (both with the spin quantum number 1/2). Let us consider Qˆ = Iˆz . We need to compute double ˆ ( − q) ], A ˆ †( − q) ] for all possible p and q (these indices should not commutators [[ Iˆz , A p p be confused with the time-dependent expectation values p(t) and q(t)) combinations (compare Table 4.2). We use the operator relations of Equation (4.40b) and perform the calculation for the case of q = 2, p = 1, corresponding to the term in Equation (4.59) with the coefficient J(ωI + ωS):   Iˆ , 1 Iˆ Sˆ  , 1 Iˆ Sˆ  = − 1  Iˆ Sˆ , Iˆ Sˆ  = 1  Iˆ Sˆ , Iˆ Sˆ  4  − − + +  4  + + − −    z 2 − −  2 + + 

(4.62)

= 21 ( Iˆ 2 − Iˆz2 )Sˆ z + 21 (Sˆ 2 − Sˆ z2 )Iˆz Evaluating the expectation value of the preceding expression, we note that 〈 Iˆ 2 − Iˆz2 〉 = 〈Sˆ 2 − Sˆ z2 〉 = 21 and obtain:   Iˆ , 1 Iˆ Sˆ  , 1 Iˆ Sˆ  =   z 2 − −  2 + + 

1 4

( Sˆ

z

+ Iˆz

)

(4.63)

Equation (4.63) tells us that the q = 2, p = 1 term in Equation (4.59), or the spectral density taken at the sum of the two Larmor frequencies J(ωI + ωS), couples the time evolution of 〈Sˆ z 〉 and 〈 Iˆz 〉 (note that we have dropped the subscript l = 2 on the symbol of spectral density). This is nothing new — the spectral density J(ωI + ωS) occurs in the expression for W2 (Equation (3.29b)),

The Redfield Relaxation Theory

87

which indeed appears in the decay terms and in the cross-relaxation terms in the Solomon equations, Equations (3.19) through Equation (3.21). The reader is encouraged to use Table 4.2 and obtain full expressions for all the rate constants in the Solomon equations. 4.2.3

Transverse Relaxation

Let us now turn to the case of transverse relaxation. The relevant operators, which give rise to detectable NMR signals and whose time evolution we wish to study, are Iˆ+ and Sˆ + . If a large difference between the Larmor frequencies is assumed, explicit calculations of the double commutators yield in this case simple exponential relaxation for both expectation values, 〈Sˆ + 〉 and 〈 Iˆ+ 〉, and no (transverse) cross-relaxation. The transverse relaxation rate for the I-spin is given by Equation (3.35) and the corresponding rate for the S-spin can be obtained by exchanging the roles of the two Larmor frequencies. When the difference between the Larmor frequencies is small — for example, for two protons with only slightly different chemical shifts — transverse relaxation should be treated with care because the secular approximation may not apply. This case is discussed in some detail in Goldman’s book (further reading). Here, we present a very brief summary. Using the interaction representation with the average Larmor frequency, ω 0, in the unperturbed Hamiltonian, one obtains that the time evolution of the density operator consists of an oscillation term and a damping. The evolution of the two relevant expectation values is described by: d ˆ I + = iδ Iˆ+ − λ Iˆ+ − µ Sˆ + dt

(4.64a)

d ˆ S+ = −iδ Sˆ + − λ Sˆ + − µ Iˆ+ dt

(4.64b)

Here, ± δ correspond to the deviations of the individual Larmor frequencies from the average frequency. The decay rate, λ, and the cross-relaxation rate, µ, are given by:

λ = π bIS2  5 J (0) + 9 J (ω 0 ) + 6 J (2ω 0 )  10

(4.65a)

which, in fact, is a limiting form of Equation (3.35) when ωI – ωS → 0, and:

µ = π bIS2  2 J (0) + 3 J (ω 0 )  5

(4.65b)

The FID corresponding to the solution of Equations (4.64) can be computed. When µ > δ, one obtains the FID corresponding to a coalesced signal, consisting of two Lorentzians with different widths and slightly different frequencies. When µ < δ, the spectrum (the Fourier transform of the FID) is a

88

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

sum of two resolved resonances with a frequency separation somewhat smaller than 2δ. When µ << δ, the effect of the transverse cross-relaxation vanishes, the secular approximation holds, and we recover the simple situation in the form of two well-separated and Lorentzian lines. Simulated lineshapes for the three cases are shown in Figure 4.3. Before leaving this section, we wish to deal with a complication (with respect to the simple case considered by Solomon) caused by presence of the scalar coupling ( J-coupling) between I and S. The unperturbed Hamiltonian needs to be modified to: Hˆ 0 = ω I Iˆz + ω SSˆ z + 2π J IS IˆzSˆ z

(4.66)

where we have assumed that the spin–spin coupling constant (2πJIS in angular frequency units) is much smaller than the difference between the Larmor frequencies. This allows us to neglect the IˆxSˆ x and IˆySˆ y terms in the coupling Hamiltonian. The corresponding spectrum now consists of four lines, in the form of two doublets with the splitting equal to 2πJIS , centered at the two Larmor frequencies. The relaxation-generating perturbation is still the dipole–dipole interaction. Under these conditions, the discussion of the longitudinal relaxation is essentially identical to the Solomon case. The relaxation of 〈Sˆ z 〉 and 〈 Iˆz 〉 is still described by a 2 × 2 relaxation matrix, with the same decay and crossrelaxation rate constants. The four lines in the spectrum correspond to four detectable (–1)-quantum coherences (see the books of Ernst et al. and Levitt (further reading)). The four coherences can be thought of as four off-diagonal elements of the density operator in the eigenstate representation, but other representations are also possible. In order to apply the Redfield approach in the operator formulation, a convenient choice is to investigate the time evolution of four expectation values: 〈Sˆ + 〉, 〈Sˆ + Iˆz 〉, 〈 Iˆ+ 〉 and 〈 Iˆ+ Sˆ z 〉 . 〈Sˆ + 〉 and 〈 Iˆ+ 〉 correspond to in-phase doublets at the Larmor frequencies of the S- and I-spins, respectively; the other two expectation

ω0 (a)

ω0 (b)

ω0 (c)

FIGURE 4.3 Simulated lineshapes, obtained from Equation (4.64a) and Equation (4.64b), corresponding to µ << δ (a), µ < δ (b), and δ < µ (c).

The Redfield Relaxation Theory

89

FIGURE 4.4 In-phase (left panel) and antiphase (right panel) doublets.

values correspond to the antiphase doublets (compare Figure 4.4). The individual lines in the spectrum can be expressed as 〈Sˆ + 〉 ± 〈2Sˆ + Iˆz 〉, and 〈 Iˆ+ 〉 ± 〈2 Iˆ+Sˆ z 〉. Also this case has been described in Goldman’s book (further reading). The result is that the in-phase coherences are not coupled to each other by relaxation, as in the case of vanishing scalar interaction. On the other hand, the individual lines within the doublets behave in a similar way as in the case of small chemical shift difference discussed earlier. The relaxation properties in a coupled two-spin system are also sensitive to the presence of anisotropic chemical shielding, another interaction leading to relaxation. We return to this issue in Section 5.2.

4.3

Relaxation in the Presence of Radiofrequency Fields

In many cases, it is useful to consider relaxation in the presence of radiofrequency fields. Several experiments are designed for such measurements and these are described in Chapter 8 through Chapter 10. We have already encountered the effects of radiofrequency fields on nuclear spin relaxation in this book in the context of the simplification of the Solomon equations through broadband decoupling of I-spins (often protons) when observing the S-spins (such as carbon-13 or nitrogen-15). The effect of the irradiation was in that case to replace the equilibrium state of a spin system by a steady state with the populations of the I-spins equalized over its energy levels. This led to single exponential S-spin relaxation and to the manifestation of cross-relaxation through the nuclear Overhauser enhancement. Here, we deal with the case in which the irradiation more directly affects the observed spins and their dynamics. Qualitatively, the key features introduced by radiofrequency (r.f.) fields can be explained as follows. First, introducing an r.f. field can change the

90

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

quantization axis of nuclear spins. One can say that spins can be locked along the new quantization axis and the effect is sometimes called spin locking. The second effect is related to cross-relaxation processes. As mentioned before, cross-relaxation between two coherences characterized by different frequencies (with the frequency difference much larger than the linewidths) is inefficient because the corresponding Redfield matrix elements are nonsecular. Conversely, cross-relaxation between the Iˆz - and Sˆ z-operators in a twospin system is secular (all populations and their combinations correspond to zero frequency) and efficient (see Figure 4.1). When the spin-locking radiofrequency field is introduced, the various magnetization components become aligned along that field and do not precess in the rotating frame defined by the first r.f. field. From the point of view of the laboratory frame, all these components precess at the same frequency and the cross-relaxation between them becomes secular. A quantitative discussion of the relaxation in the presence of radiofrequency fields can be found in several reviews7,12–17 and the following presentation is based on that work, in particular the paper by Goldman.7 Consider a heteronuclear system of two spins, I and S, without scalar coupling in a static magnetic field B0. The spins are subjected to a linearly polarized radiofrequency field perpendicular to the static field, with the frequency ω, in the vicinity of the Larmor frequency of the S-spins and with the amplitude B1 (ω1 = − γSB1 in the angular frequency units). We assume that the radiofrequency field is weak compared to the static field, B1 << B0, but strong enough to fulfill the relation Rrelax << ω1, where Rrelax is an arbitrary relaxation matrix element. The linearly polarized r.f. field can be expressed as a sum of circularly polarized r.f. fields rotating in opposite directions, with frequencies + ω and − ω (see, for example, Slichter in further reading). One of the rotating components will be very far from ωS = −γSB0 and it is sufficient to retain the one close to the Larmor frequency, let us say, +ω. In the same way, the rotating field is assumed not to interact with the I-spin. The unperturbed Hamiltonian now contains the Zeeman interactions between both spins and the static field, as well as between the spin S and the rotating field: Hˆ 0 = ω I Iˆz + ω SSˆ z + ω 1 (Sˆ x cos ω t + Sˆ y sin ω t) = ω I Iˆz + ω SSˆ z + ω 1 exp(−iω Sˆ zt)Sˆ x exp(iω Sˆ zt)

(4.67)

= exp(−iω Sˆ zt)(ω I Iˆz + ω SSˆ z + ω 1Sˆ x )exp(iω Sˆ zt) In the second line, we have expressed the rotation around the z-axis using exponential operators as previously described, and in the third line we have exploited the fact that rotation around the z-axis leaves the operators Iˆz and Sˆ z unchanged. In addition to the Hˆ 0 term, we have the perturbation, Hˆ 1 (t), causing relaxation.

The Redfield Relaxation Theory

91

Following the procedure of Section 4.1, we want to transform all operators to the interaction representation. We choose to do it in two steps. In the first step, we transform to the frame rotating at the frequency ω around the z direction of the static field, introduced in Section 1.3. This amounts to the transformation of operators according to:  Qˆ → Qˆ = exp(iω Sˆ zt)Qˆ exp(−iω Sˆ zt)

(4.68)

By a calculation similar to Equation (4.5), we obtain the Liouville–von Neumann equation of motion for the density operator in this interaction representation: d ˆ ρ(t) = i  ρˆ (t), Hˆ eff + ω I Iˆz  dt

(4.69)

with the effective S-spin Hamiltonian: ˆ eff = (ω S − ω )Sˆ z + ω 1Sˆ x = ∆Sˆ z + ω 1Sˆ x = ΩSˆ Z H

(4.70)

In the second equality, we have introduced the offset frequency, ∆ = ωS – ω, characterizing the off-resonance condition of the applied radiofrequency. As indicated by the last part of Equation (4.70), the effective Hamiltonian is a Zeeman Hamiltonian with a static field along the 0Z direction, which is tilted with respect to the laboratory z-axis by an angle, Θ, so that: tan Θ =

ω1 ∆

(4.71a)

The relation between the laboratory frame and the tilted rotating frame is illustrated in Figure 4.5. The Larmor frequency in the new frame is: Ω = (ω 12 + ∆ 2 )

1/2

(4.71b)

FIGURE 4.5 Illustration of the relation between the laboratory frame and the tilted rotating frame.

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Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

The perturbation Hamiltonian, in the form of the dipolar IS interaction, needs also to be transformed to the tilted rotating frame, in the same way as we did with the density operator:  Hˆ 1 (t) = exp(iω Sˆ zt)Hˆ 1 (t)exp(−iω Sˆ zt)

(4.72)

In the second step, we go over to a doubly rotating frame by:

(

)

(

   Qˆ → Q = exp i(ΩSˆ Z + ω I Iˆz )t Qˆ exp −i(ΩSˆ Z + ω I Iˆz )t

)

(4.73)

 For the consistency of notation, the symbol Q should also contain an operator “hat.” This would, however, be awkward and we hope that the notation is not confusing in any way. In this interaction representation, we can derive a master equation analogous to Equation (4.26): ∞

dρ     = −   ρ (t), H 1 (t + τ )  , H 1 (t)  dτ   dt   0

∫

(4.74)

Equation (4.74) or its operator counterpart (compare Goldman7) needs a correction for the finite temperature of the lattice, as discussed earlier. Using the perturbation Hamiltonian in the form of Equation (4.37a) and Equation (4.37b), the relaxation properties of the system can be evaluated in a reasonably straightforward way and the results can be found in the review by Desvaux and Barthault.16 Here, we only discuss the results for some limiting cases. The first case of interest is an experiment in which we perform on-resonance irradiation of the S-spin. In this case, ∆ = 0, Θ = π/2, and Ω = ω1. This condition leads to a situation in which the expectation value of the Z-component of the S magnetization, 〈Sˆ Z 〉 , decays exponentially with a time constant denoted T1ρ (which stands for T1 in the rotating frame). The expression for the corresponding rate is: T1−ρ1 = π bIS2 [ 4 J (ω 1 ) + J (ω I − ω S ) + 3 J (ω S ) + 6 J (ω I ) + 6J (ω I + ω S )] 10

(4.75)

which is almost identical to the expression for T2–1, given for the specific case of the carbon-13–proton system, in Equation (3.35). The difference lies in the fact that the zero-frequency spectral density is replaced by the spectral density taken at the frequency ω1. Because we assumed B1 << B0, we then have ω1 << ωS and the difference between T1ρ–1 and T2–1 can be neglected in the absence of very slow motions. The difference can in fact be exploited to probe very slow motions and we will

The Redfield Relaxation Theory

93

return to this point in the discussion of experimental methods in Chapter 13. The close similarity of the two relaxation rates is a consequence of the fact that in the case of on-resonance irradiation, the quantization axis for the Ispin is changed to the transverse plane. In other words, 〈Sˆ Z 〉 becomes identical to the transverse magnetization, 〈Sˆ x 〉 or 〈Sˆ y 〉 . A second interesting situation arises if we apply the radiofrequency off-resonance, with ∆ ≠ 0, 0 < Θ < π/2, and Ω > ω1. Physically, this case corresponds to a measurement of the relaxation of a magnetization component between the longitudinal (with the decay rate T1–1, assuming that the I-spin is rapidly relaxing or saturated) and the transverse magnetization. The relaxation rate, (T1offρ − res )−1, in this case is given by: (T1offρ − res )−1 = π bIS2 [ 4 sin 2 ΘJ (ω 1 ) + (2 − sin 2 Θ)) J (ω I − ω S ) 10 + (6 − 3 sin 2 Θ) J (ω S ) + 6 sin 2 ΘJ (ω I ) + 6(2 − sin 2 Θ) J (ω I + ω S )] (4.76) It is easy to see that Equation (4.76) simplifies to Equation (4.75) if Θ = π/2 and to Equation (3.31a) when Θ = 0 (the reader is encouraged to check this statement). The final limiting case is that of a homonuclear IS-spin system, with the two spins (e.g., two protons) with different chemical shifts, but with the radiofrequency field strong enough to fulfill ∆ << ω1 and Θ ≈ π/2 for both spins, i.e., to change the quantization axes of both spins to the transverse plane of the laboratory frame. The term ω I Iˆz in Equation (4.69) is then included in the effective Hamiltonian, which becomes: Hˆ eff = (ω I − ω )Iˆz + (ω S − ω )Sˆ z + ω 1 ( Iˆx + Sˆ x ) = Ω I IˆZ + Ω SSˆ Z

(4.77)

where the Larmor frequencies for the two spins are allowed to be different in the laboratory frame and in the common tilted rotating frame with its unique Z-axis. The consequence of both spins being locked along the common axis in the rotating frame is that the cross-relaxation between the two Z-components becomes secular. In other words, the relaxation of the two Z-components (which indeed are transverse components in the laboratory frame) becomes a coupled process, following an equation system similar to the Solomon case. Bothner-By17 gave the following equations for an isolated spin pair:

−

 d dt  

 IˆZ   R σ ISROE   2I =    ROE R2 S   Sˆ Z   σ IS 

IˆZ   ˆS  Z 

(4.78)

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Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

with R2 I = R2 S = π bIS2 [5 J (0) + 9 J (ω I ) + 6 J (2ω I )] 10

(4.79a)

σ ISROE = π bIS2 [ 4 J (0) + 6 J (ω I )] 10

(4.79b)

and

The transverse relaxation rates, R2I and R2S, are identical to Equation (4.75) at the limit of vanishing frequency difference between the I- and S-spins and ω1 approximated by zero. They are also identical to the relaxation rate, λ, introduced in the context of transverse relaxation in systems with small chemical shift difference. The transverse cross-relaxation rate, σ ISROE , is the same as µ introduced in Equation (4.65b). It is an important quantity. As opposed to the longitudinal cross-relaxation rate (compare Equation (3.22c) and Equations (3.29)), which can be both positive and negative and crosses zero at a certain value of the correlation time, the transverse rate is always positive. This has important consequences for determining which nuclei are spatially close to each other, and thus determination of structure, in mediumsize molecules, which often fall in the regime of having close-to-zero longitudinal cross-relaxation rates. We will return to the experimental procedures for determining the transverse cross-relaxation rates in Chapter 9.

References 1. Wangsness, R.K. and Bloch, F. The dynamic theory of nuclear induction, Phys. Rev., 89, 728–739, 1953. 2. Redfield, A.G. The theory of relaxation processes, Adv. Magn. Reson., 1, 1–32, 1965. 3. Redfield, A.G. Relaxation theory: density matrix formulation, in Encyclopedia of Nuclear Magnetic Resonance, Grant, D.M. and Harris, R.K., Eds. (Chichester: Wiley), 4085–4092, 1996. 4. Hubbard, P.S. Quantum-mechanical and semiclassical forms of the density operator theory of relaxation, Rev. Mod. Phys., 33, 249–264, 1961. 5. Mayne, C.L. and Smith, S.A. Relaxation processes in coupled-spin systems, in Encyclopedia of Nuclear Magnetic Resonance, Grant, D.M. and Harris, R.K., Eds. (Chichester: Wiley), 4053–4071, 1996. 6. Werbelow, L.G. and Kowalewski, J. Nuclear spin relaxation of spin one-half nuclei in the presence of neighboring higher-spin nuclei, J. Chem. Phys., 107, 2775–2781, 1997. 7. Goldman, M. Formal theory of spin-lattice relaxation, J. Magn. Reson., 149, 160–187, 2001. 8. Murali, N. and Krishnan, V. A primer for nuclear magnetic relaxation in liquids, Concepts Magn. Reson. Part A, 17A, 86–116, 2003.

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9. Hubbard, P.S. Some properties of correlation functions of irreducible tensor operators, Phys. Rev., 180, 319–326, 1969. 10. Werbelow, L. and London, R.E. Dynamic frequency shift, Concepts Magn. Reson., 8, 325–338, 1996. 11. Werbelow, L.G. Dynamic frequency shift, in Encyclopedia of Nuclear Magnetic Resonance, Grant, D.M. and Harris, R.K., Eds. (Chichester: Wiley), 1776–1783, 1996. 12. Bull, T.E. Relaxation in the rotating frame in liquids, Prog. Nucl. Magn. Reson. Spect., 24, 377–410, 1992. 13. Bax, A. and Grzesiek, S. ROESY, in Encyclopedia of Nuclear Magnetic Resonance, Grant, D.M. and Harris, R.K., Eds. (Chichester: Wiley), 4157–4167, 1996. 14. Schleich, T. Rotating frame spin-lattice relaxation off-resonance, in Encyclopedia of Nuclear Magnetic Resonance, Grant, D.M. and Harris, R.K., Eds. (Chichester: Wiley), 4167–4174, 1996. 15. Palke, W.E. and Gerig, J.T. Relaxation in the presence of an r.f. field, Concept Magn. Reson., 9, 347–353, 1997. 16. Desvaux, H. and Berthault, P. Study of dynamic processes in liquids using off-resonance r.f. irradiation, Prog. NMR Spectr., 35, 295–340, 1999. 17. Bothner–By, A.A., Stephens, R.L., and Lee, J.M. Structure determination of a tetrasaccharide: transient nuclear Overhauser effects in the rotating frame, J. Am. Chem. Soc., 106, 811–813, 1984.

5 Applications of Redfield Theory to Systems of Spin 1/2 Nuclei

CONTENTS 5.1 Multispin Dipolar Relaxation: Magnetization Modes ............................................................................................................98 5.1.1 Magnetization Modes .....................................................................99 5.1.2 Relaxation Matrix for Magnetization Modes ...........................102 5.2 Anisotropic Chemical Shielding and its Relaxation Effects.......................................................................................105 5.2.1 The Chemical Shielding Tensor ..................................................105 5.2.2 The CSA Hamiltonian ..................................................................108 5.2.3 The CSA Relaxation ...................................................................... 112 5.2.4 The DD/CSA Interference ........................................................... 115 5.2.5 Transverse Relaxation Effects of the DD/CSA Interference..................................................................................... 118 5.2.6 The Antisymmetric Component .................................................120 5.3 Spin-Rotation Relaxation..........................................................................121 5.4 Other Relaxation Mechanisms for Spin 1/2 Nuclei..........................................................................................123 References ............................................................................................. 124

In this chapter, we present applications and results of the Redfield theory. We restrict discussion to systems of spin 1/2 nuclei; nuclei with spin 1 or higher (quadrupolar nuclei) will be the subject of Chapter 14. So far, we have only been concerned with dipolar relaxation involving two coupled spins. However, other interactions also vary with time through stochastic processes related to molecular motion. We begin this chapter by looking at some features of dipolar relaxation in systems consisting of more than two spins and then proceed to deal with relaxation effects originating from other interactions involving spin 1/2 nuclei. In particular, relaxation through the anisotropic chemical shift is important in many spin 1/2 systems.

97

98

5.1

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

Multispin Dipolar Relaxation: Magnetization Modes

An interesting complication of the Solomon-like description arises when we have a spin system consisting of more than two spins (all with the spin quantum number of 1/2) with mutual dipolar interactions. The unperturbed Hamiltonian consists of the Zeeman interactions between all the spins and the magnetic field and, possibly, the scalar spin–spin coupling. The perturbation is a sum of dipole–dipole interactions. The summation over h, running over all pairs of dipolar nuclei, leads to a generalization of Equation (4.37a):

Hˆ 1 (t) =

∑

Hˆ η (t) =

η

l

∑ ∑ (−1) V ξη

η

q

η l ,q

(t)Tˆlη,− q

(5.1)

q=− l

We can use this expression for the dipole–dipole Hamiltonian to compute the spectral density Jaa ′bb′(w), using the formulation given in Section 4.1 by substituting this form of the perturbation Hamiltonian into Equation (4.29). The spectral density thus becomes: ∞

Jαα ′ ββ ′ (ω ) = 2

∫ α Hˆ (t) α ′

β ′ Hˆ 1 (t + τ ) β exp(−iωτ )dτ

1

0

=2

∑ ∑ ξ ξ ∑ ∑ (−1) η η′

η

η'

q

q + q′

α Tˆlη,− q α ′ β ′ Tˆl†,−ηq′′ β

q'

(5.2)

∞

∫V

η l ,q

×

(t)Vl*,qη′′ (t + τ ) exp(−iωτ )dτ

0

=

∑ ∑ ξ ξ ∑ α Tˆ

η l ,− q

η η′

η

η′

α ′ β ′ Tˆl†,−ηq′′ β J lηη′ (ω )

q

We now deal with a Hamiltonian that contains several terms and the summation runs over all possible h, h′ indices; thus, we obtain “cross-terms” between different components of the Hamiltonian. The quantity * η′ η ′ Glηη , q , q′ (τ ) = 〈Vl , q (t)Vl , q′ (t + τ )〉 for h = h′ is our familiar time autocorrelation function and its counterpart for h ≠ h′ is called the cross-correlation function. It describes the correlation between a classical random function contained in the dipolar Hamiltonian Hˆ η (t) and its counterpart related to Hˆ η′ (t + τ ). The ηη′ ′ Fourier transform of Glηη , q , q′ (τ ), denoted J l (ω ), is called the cross-correlation spectral density. In the last equality in Equation (5.2), we have used the fact

Applications of Redfield Theory to Systems of Spin 1/2 Nuclei

99

that the symmetry relation of Equation (4.52) also applies to the crosscorrelation functions.

5.1.1

Magnetization Modes

The cross-correlation spectral densities do influence relaxation properties in multispin systems, at least under certain conditions. These effects have been the subject of reviews by Kumar and coworkers1 as well as other authors.2,3 Equation (4.58) or Equation (4.61) can be generalized to accommodate the more complicated Hˆ 1 (t) of Equation (5.1), but this becomes rather messy. Therefore, we will take a different approach. We will examine only cross-correlation effects in longitudinal relaxation because this is a bit easier than transverse relaxation. We are going to introduce the multispin effects in longitudinal relaxation (evolution of diagonal elements of the density matrix) using the concept of magnetization modes. This approach to describe multispin relaxation phenomena has been discussed extensively by Werbelow and Grant4 and Canet.5 We use the expressions for the time evolution of the density matrix element, Equation (4.30a) or Equation (4.33), as a starting point. The elements of the relaxation supermatrix were obtained in the previous chapter as linear combinations of the spectral density functions at different frequencies (Equation 4.28). For a multispin system, we need to use this formulation in combination with Equation (5.2). We then introduce a new operator basis, in the form of linear combinations of the |a〉 〈a | operators, according to: vˆ i =

∑Q

i ,α

α α

(5.3)

α

The matrix Q with elements Qi,a describes the transformation between the two basis sets. The density operator can be expanded in the new basis, according to:

ρˆ =

∑ c vˆ = ∑ ∑ Q

i ,α

i i

i

ραα vˆ i

(5.4a)

α

i

That is, the expansion coefficients, ci, for the density operator in the new basis can be obtained through: ci =

∑Q

i ,α

ραα

(5.4b)

α

Using this basis set and applying the correction for finite temperature, Equation (4.30a) or Equation (4.33) can be written as: d vˆ i (t) = − dt

∑ Γ ( vˆ ij

j

j

(t) − vˆ j

eq

)

(5.5)

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Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

where Γij are the relaxation supermatrix elements in the new basis: − Γ ij =

∑Q

i ,α

(5.6)

Rααββ Qj ,β

α ,β

We will examine one example of this kind of analysis, following the presentation by Werbelow and Grant.4 In the example, the system consists of three different spin 1/2 nuclei, denoted “AMX” in the common notation. We assume that the three spins are coupled through mutual scalar couplings, denoted by JAM, JAX, and JMX and that there are no degeneracies in the sense that each line in the spectrum corresponds to a single transition between eigenstates of Hˆ 0 . In addition, we assume that all the coupling constants are much smaller than the differences in chemical shifts (expressed in common units), the so-called weak coupling limit. The NMR spectrum of such a spin system consists of three groups of signals; each group is a symmetric doublet of doublets (compare Figure 5.1). We begin the analysis by setting up a new, useful form of the vector basis and then discuss its properties and its relation to the spectrum in Figure 5.1. The unperturbed Hamiltonian can be expressed in analogy with Equation (4.66) by adding one more Zeeman term and two more scalar coupling terms. The eigenvectors are characterized by the three quantum numbers mAmMmX〉: 1 ≡ +++ , 2 ≡ ++− , 3 ≡ +−+ , 4 ≡ −++ , 5 ≡ +−− , 6 ≡ −+− , 7 ≡ −−+ , 8 ≡ −−−

(5.7)

where the shorthand notation +++〉 implies mA = +1/2, mM = +1/2, and mX = +1/2. We perform the transformation of Equation (5.3) or Equations (5.4) in two

ω1

ω2

ω3

FIGURE 5.1 The AMX spectrum. The spectrum corresponds to the case J12 > J23 > J13.

Applications of Redfield Theory to Systems of Spin 1/2 Nuclei

101

steps. It is useful to combine the ket-bra products in such a way that the resulting operators have definite symmetry properties with respect to spin inversion:  a vˆ 1′ a  vˆ 2′ a  vˆ 3′ aˆ  v4′  s vˆ 5′   s vˆ 6′ s  vˆ 7′ s  vˆ 8′

             =               

1 1− 8 8   2 2 − 7 7  3 3 − 6 6  4 4 − 5 5  1 1 + 8 8  2 2 + 7 7   3 3 + 6 6  4 4 + 5 5 

(5.8)

Spin inversion refers to changing the sign of all the magnetic quantum numbers. Clearly, the first four of the new vectors are antisymmetric with respect to spin inversion — i.e., they change sign as a result of this operation — and the last four vectors are symmetric. We indicate these symmetry properties by the superscripts a and s. In the second step, we perform a transformation of the vectors v′ into v, in such a way that the spin-inversion symmetry is retained:  a vˆ 1  1 1 1 −1 a    vˆ 2  1 1 −1 1 a   ˆ  v3  1 −1 1 1 aˆ   1 v  4  = 1 −1 −1 −1  s vˆ 1  2 0 0 0 0     s vˆ 2  0 0 0 0 s   ˆ  v3  0 0 0 0 sˆ   v  4  0 0 0 0

0 0 0 0 0 0 0 0 1 1 1 −1 1 −1 1 1

0 0  a vˆ 1′    0 0  a vˆ 2′    0 0  a vˆ 3′  0 0  a vˆ 4′  −1 −1  s vˆ 5′    1 −1  s vˆ 6′    −1 1  s vˆ 7′    1 1  s vˆ 8′ 

(5.9)

This transformation has the advantage that all the elements of v have a straightforward physical interpretation. The magnetization modes are the expectation values of the new operator set. The first three magnetization modes correspond simply to the expectation values of the Iˆz operators for the three spins: a

a

a

v1 (t) = Tr  ρˆ (t)IˆzA 

(5.10a)

v2 (t) = Tr  ρˆ (t)IˆzM   

(5.10b)

v3 (t) = Tr  ρˆ (t)IˆzX   

(5.10c)

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Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

Note that the expectation values are explicitly time dependent, but the operators vˆ i are not. The reader is advised to check that Relations (5.10) indeed are fulfilled. Physically, after subtraction of the corresponding equilibrium value, the magnetization mode av1(t) represents the summed deviations from thermal equilibrium of the four lines in the multiplet centered at the resonance frequency of the A spins. The modes av2(t) and av3(t) have the same meaning for the multiplets at the resonance frequency of M and X, respectively. The fourth element, av4(t), can also be related to a trace of an operator expression: a

v1 (t) = Tr  ρˆ (t) 4 IˆzA IˆzM IˆzX   

(5.11)

The relation between the product operator 4 IˆzA IˆzM IˆzX and the spectrum is less apparent. Still, its expectation value is again a measurable quantity: the deviation from thermal equilibrium for the sum of the outer lines minus the sum of the inner lines (or vice versa, depending on the signs of the scalar coupling constants) for each of the multiplets. We can say that av4(t) describes one kind of multiplet asymmetry: the difference in the evolution of the individual lines in a multiplet split by the scalar coupling. The first three symmetric magnetization modes, sv1(t), sv2(t), and sv3(t), can also be related to the expectation values of product operators: v1 (t) = Tr  ρˆ (t)2 IˆzA IˆzM   

(5.12a)

v2 (t) = Tr  ρˆ (t)2 IˆzA IˆzX   

(5.12b)

v3 (t) = Tr  ρˆ (t)2 IˆzM IˆzX   

(5.12c)

s

s

s

and correspond to other multiplet asymmetries in the AMX spectrum. The last symmetric element, s v4 (t) = Tr [ ρˆ (t) 21 Eˆ ], corresponds to the trace of the density operator multiplied with the identity operator or to the sum of the populations of all spin states, which is an invariant quantity.

5.1.2

Relaxation Matrix for Magnetization Modes

Having defined the magnetization modes and their expectation values, we now turn to the relaxation matrix in the magnetization mode basis, Γ. We assume that the three dipolar interactions (AM, AX, and MX) are the only mechanism of relaxation. The calculation of the relaxation matrix elements is tedious and we will here provide only a brief, general discussion of their

Applications of Redfield Theory to Systems of Spin 1/2 Nuclei

103

properties and define some selected, important expressions. For details, the reader is referred to the review of Werbelow and Grant4 and the original references quoted there. The first important property of the relaxation matrix is that, if only the dipolar interactions are considered, then the asymmetric and the symmetric manifolds are not connected by the relaxation matrix. We can thus write:  aΓ d  a v(t) s  = − dt  v(t)  0

0   a v(t) − a v eq    Γ   s v(t) − s v eq 

s

(5.13)

The second important property is that the relaxation matrix is a symmetric matrix, Γij = Γji. The diagonal elements, Γii, corresponding to the three first antisymmetric magnetization modes are important. These are obtained from a straightforward generalization of the expression for rS = T1–1 in Equation (3.31a). In this basis, aΓ11, for example, becomes: a

Γ 11 = ρ AM + ρ AX

(5.14a)

with 2  J AM , AM (ω A − ω M ) + 3 J AM , AM (ω A ) + 6 J AM , AM (ω A + ω M )  (5.14b) ρ AM = π bAM   5

The subscript AM,AM on the spectral density explicitly indicates the autocorrelation of the AM dipolar interaction. Analogous expressions for aΓ22,aΓ33 can readily be obtained mutatis mutandis. Likewise, the off-diagonal elements aΓ = s a a 12 AM, Γ13 = sAX, and Γ23 = sMX (and their symmetry-related counterparts) are cross-relaxation rates, fully analogous to the expressions given in Equation (3.22c) and Equations (3.29). These terms couple together the magnetization modes given by the expectation values of the Iˆz operators. The elements aΓ14, aΓ24, and aΓ34 are qualitatively different and interesting. As opposed to the other elements in the antisymmetric part of the relaxation matrix, which was seen to contain only the autocorrelated spectral densities, the matrix elements aΓ14, aΓ24, and aΓ34, coupling together the Iˆz terms and the 4 IˆzA IˆzM IˆzX, only depend on the cross-correlated spectral densities: a

Γ 14 = δ MAX = 6π bAMbAX J AM , AX (ω A ) = 6π bAMbAX K MAX (ω A ) 5 5

(5.15)

For that reason, we will call these relaxation matrix elements the crosscorrelated relaxation rates (CCRRs), which is the standard nomenclature. The notation “cross-correlated cross-relaxation rates” might be more appropriate, but is judged to be a bit too clumsy. The cross-correlated dipolar spectral densities are denoted JAM,AX(w) or KMAX(w); both symbols are commonly used

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Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

in the literature. We will return to calculations of the cross-correlation functions and corresponding spectral densities in Chapter 6. The matrix elements for the part of the relaxation matrix corresponding to the symmetric modes are given as combinations of various r, s, and d terms (compare Werbelow and Grant4). Thus, we find that the longitudinal relaxation in AMX systems is in general multiexponential, which indeed is similar to the two-spin case described by the Solomon equations. Starting with these equations, we were able to recover a single, well-defined T1 under some limiting conditions. One was the case of I and S being identical spins. The three-spin counterparts of this situation, the A3 or AX2 spin systems, are more complicated. The A3 system was treated by Hubbard as early as 19586 and both these systems, along with several other cases, were discussed in large detail by Werbelow and Grant.4 We refer the reader to that work for a comprehensive presentation. Another limiting situation, resulting in a single exponential longitudinal relaxation in a two-spin system, was to apply the experimental trick of spin decoupling, i.e., saturation of all transitions for one of the spin, I. The analogue of this experiment can easily be realized in a heteronuclear three-spin system, where the A spin represents a nuclear species other than M and X. A common case of experimental interest occurs when A = 13C, while M and X are nonequivalent protons. As discussed by Werbelow and Grant4 and by Brondeau and coworkers,7 the broadband decoupling of protons results in a single exponential carbon relaxation. The 13C T1–1 is given by Equation (5.14a) and Equation (5.14b), which imply that the relaxation contributions from the individual carbonproton dipolar interactions are additive. The phenomenon of the nuclear Overhauser enhancement arises also in the form of a simple linear combination of two separate enhancements characteristic for the two-spin system. In the simple case of a methylene carbon carrying two magnetically nonequivalent protons with identical CH distances, the relaxation rate expression in Equation (5.14a) gives the result that the equation for longitudinal relaxation in Equation (3.31a) has only to be multiplied by two to be valid. Equation (3.31b) for the NOE retains its validity also for this case. Similar results are obtained for a carbon-13 nucleus interacting with more complicated proton spin systems, as long as the protons are nonequivalent. In the cases in which magnetic equivalence is present, the situation becomes more complicated and the cross-correlation effects may not be completely negligible, even under broadband irradiation of the proton spins. The reader is referred to Werbelow and Grant4 for a general discussion and to Kowalewski et al.8 for a careful case study. Summarizing the longitudinal relaxation in the AMX case, we have found that one should exercise a certain care when selecting the basis set in which the relaxation matrix is to be evaluated. The set of magnetization modes is often a good choice. Using this formulation, we recover a three-spin analogue of the Solomon equations, if we can neglect the cross-correlation effects between pairs of dipolar interactions. We will return to the experimental

Applications of Redfield Theory to Systems of Spin 1/2 Nuclei

105

demonstration of the role of dipolar cross-correlations in Chapter 10. Spin decoupling in a heteronuclear spin system can simplify the picture, but caution is recommended in case of the presence of magnetic equivalence.

5.2

Anisotropic Chemical Shielding and its Relaxation Effects

The dipole–dipole interaction is often the most important source of spin relaxation for I = 1/2 spins, but not the only one. Another important interaction is the anisotropic chemical shift or the chemical shielding anisotropy (CSA). The CSA as a relaxation mechanism has been discussed thoroughly in several reviews, e.g., by Smith et al.9–11 and Anet and O’Leary.12,13

5.2.1

The Chemical Shielding Tensor

In the presence of chemical shielding, the local magnetic field at the nucleus is not equal to the external field of the NMR magnet, but is given by: Bloc = B0 (1 − σ ) = B0 − Bind

(5.16)

with the induced field, Bind = sB0. The shielding is in general anisotropic, meaning that it is orientation dependent. This means that, for a powder sample, the shielding experienced by different crystallites in the sample is different, depending on their orientation with respect to the magnetic field. The chemical shielding is described as a tensor property of rank 2. The symbol s denotes this shielding tensor. We can express the shielding tensor in a Cartesian form: σ xx σ xy σ xz    σ = σ yx σ yy σ yz     σ zx σ zy σ zz 

(5.17)

The shielding tensor is a property of a particular nucleus in a molecule or a crystal and is therefore naturally expressed in the molecular coordinate system. The meaning of the symbol sxy can be understood by recognizing that sxyB0 is the induced field along the molecular x-axis caused by an external field B0 oriented along the molecular y-axis. The necessity of describing the phenomenon of shielding with a shielding tensor, rather than a simpler object such as a scalar, reflects the fact that the induced field does not need to be parallel to the external field. The shielding tensor in the

106

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

Cartesian form does not need to be symmetric: sxy ≠ syx, which is an obvious consequence of the meaning of the tensor component. We thus have a set of nine independent elements. The Cartesian form of the shielding tensor is not necessarily the most useful one. We can rewrite the Cartesian rank-2 tensor as a sum of three terms: σ xx  σ yx   σ zx

 1 0 0  σ xy σ xz  0   1  σ yy σ yz  = σ iso  0 1 0 +  −σ xy + σ yx   2  σ zy σ zz   0 0 1  −σ xz + σ zx  σ xx − σ iso  +  21 σ xy + σ yx   1 σ xz + σ zx 2

( (

1 2

(σ

) σ ) (σ

xy

yy

1 2

+ σ yx

− σ iso

yz

+ σ zy

σ xy − σ yx σ xz − σ zx   0 σ yz − σ zy   −σ yz + σ zy 0 

) (σ (σ ) σ 1 2

1 2

) )

+ σ zx    yz + σ zy   − σ zz iso  xz

= σ ( 0) + σ (1) + σ ( 2 ) (5.18) with σ iso = 13 (σ xx + σ yy + σ zz ). The first term, s(0), is a scalar multiplied by a unit tensor. The second term, s(1), is an antisymmetric tensor, and the third, s(2), is a symmetric and traceless tensor. The symmetric part of the shielding tensor may be transformed into a diagonal form — i.e., into a form in which only the diagonal elements are nonzero — by a suitable rotation of the coordinate system. We call this process the diagonalization of the tensor. The new frame, in which the symmetric part of the shielding tensor is diagonal, is called the principal frame or the principal axis system (PAS) of the tensor. In the principal frame, the traceless, symmetric, rank-2 tensor becomes:

σ

(2)

PAS σ xx − σ iso  = 0  0 

σ

PAS yy

0 − σ iso 0

 − 1 (1 − η)  0 0 0   2  =δ 0 0 − 21 (1 + η) 0     σ zzPAS − σ isoo   0 0 1  (5.19)

where PAS PAS δ = σ zzPAS − σ iso = 23 [σ zzPAS − (σ xx + σ yy )/2] = 23 ∆σ

(5.20a)

PAS PAS PAS PAS η = (σ xx − σ yy )/δ = 3(σ xx − σ yy )/2∆σ

(5.20b)

and

Applications of Redfield Theory to Systems of Spin 1/2 Nuclei

107

We should remember that the rotation of the coordinate system leaves the trace of the tensor unchanged. Thus, in agreement with the properties of a scalar, siso does not change when we go over from an arbitrary initial frame to PAS PAS the PAS. The tensor components σ xx , σ yy and σ zzPAS are denoted as principal components of the shielding tensor. The labels x, y, and z follow the convention: PAS PAS σ zzPAS − σ iso ≥ σ xx − σ iso ≥ σ yy − σ iso

(5.21)

∆s is called the chemical shielding anisotropy and h the asymmetry parameter. PAS PAS = σ yy = σ ⊥ , we say that the When the asymmetry parameter vanishes, σ xx shielding tensor is axially symmetric (but still anisotropic; the shielding PAS PAS becomes isotropic only if σ xx = σ yy = σ zzPAS). The symmetry axis is the z-axis, according to the convention in Equation (5.21), and one often uses the notation σ zzPAS = s|| . It is important to realize that the symmetry properties of the shielding tensor are closely related to the local symmetry of the molecular framework around the nucleus under consideration. Referring to Anet and O’Leary12 for a more detailed presentation, we wish here to mention a single example of these relations. The carbon-13 nucleus in chloroform resides on a threefold symmetry axis of the molecule, which makes the shielding tensor axially symmetric and lets us identify the molecular symmetry axis with the z-axis of the tensor. It is illustrative to relate the principal components of the shielding tensor to the spectrum of a solid powder sample analogous to the dipolar powder pattern in Figure 3.4. Such spectra for the axial and nonaxial case are shown in Figure 5.2. In the axial case, the shielding anisotropy simplifies to ∆s = s|| – s⊥ and corresponds to the width of the powder pattern. The powder pattern is a superposition of signals corresponding to different orientations of the molecule/microcrystallite with respect to the magnetic field. Each orientation corresponds, in turn, to a slightly different local field felt by the nucleus. The random tumbling of a molecule in solution results thus in a randomly varying local field at the nucleus. This is reminiscent of the simple example considered in Chapter 2 and provides a qualitative explanation of the role of CSA as a relaxation mechanism. Note that all the discussion and terminology introduced following Equation (5.19) refers to the symmetric part of the shielding tensor. Thus, the concepts of anisotropy of the tensor and its asymmetry parameter have nothing to do with the antisymmetric part, s(1), of the shielding tensor. The choice of the axis system that makes the symmetric part diagonal does not make the antisymmetric part vanish. As discussed in the reviews by Anet and O’Leary12,13 and in references quoted there, the antisymmetric part of the shielding tensor always results in an induced field perpendicular to the external field, with the magnitude dependent

108

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

−150 −100 −50

0

−150 −100 −50

0

50

100

150

50

100

150

ω(ppm) (a)

ω(ppm) (b)

200

200

250

250

FIGURE 5.2 Illustration of the appearance of the powder pattern due to the CSA tensor for an axially symmetric case (a) and a nonaxial case (b). Simulations were performed using s⊥ = –60 ppm and s|| = + 140 ppm (a), and syy = –60 ppm, sxx = 0 ppm, and szz = + 140 ppm (b).

on the molecular orientation. The antisymmetric shielding tensor contributes to the solid-state powder pattern as a second-order effect. From the point of view of relaxation, it acts essentially as an independent source of relaxation (relaxation mechanism), usually much less important than the “ordinary” CSA mechanism connected to the symmetric rank-2 tensor. 5.2.2

The CSA Hamiltonian

In order to formulate the theory of the CSA relaxation in quantitative terms, we need to formulate the chemical shielding Hamiltonian and, eventually, turn it into an expression in terms of irreducible tensors. The Hamiltonian for a single spin in the Cartesian coordinates is: Hˆ ICS = γ I Iˆ ⋅ σ ⋅ B

(5.22)

Applications of Redfield Theory to Systems of Spin 1/2 Nuclei

109

We also introduce the irreducible spherical components of the shielding tensor9 according to:

σ 0 ,0 = − 3σ iso i

σ 1, 0 = −

2

(5.23a)

(σ xy − σ yx )

(

(5.23b)

)

σ 1,±1 = − 21 σ zx − σ xz ± i σ zy − σ yz  σ 2 ,0 =

3 2

(σ zz − σ iso )

(5.23c)

(5.23d)

(

)

(5.23e)

(

)

(5.23f)

σ 2 ,±1 = ∓ 21 σ xz + σ zx ± i σ yz + σ zy  σ 2 ,±2 = 21 σ xx − σ yy ± i σ xy + σ yx 

We can easily recognize simple relations between the spherical components and the elements of s(0), s(1), and s(2) in Equation (5.18). We concentrate for the moment on the rank-2 spherical components, which have important implications for relaxation. These obtain a simpler form when expressed in the principal axis system:

σ 2PAS ,0 =

3 2

δ=

2 3

∆σ

(5.24a)

σ 2PAS ,±1 = 0

(5.24b)

1 σ 2PAS ,±2 = 2 δη

(5.24c)

Given the simplicity of the preceding expressions, it is advantageous to use the PAS as a starting point for the discussion of the rank-2 shielding tensor and the CSA relaxation. This discussion involves a series of coordinate frame transformations, which are conveniently carried out using the basic properties of the spherical tensors (see Brink and Satchler in the further reading section in the Preface for more details). Using these properties, we can express the shielding tensor in the arbitrary axis system (AAS) as: 2

σ 2AAS ,q =

∑D

2 p ,q

p=−2

(Ω PAS, AAS ) σ 2PAS ,p

(5.25)

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Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

The symbol ΩPAS,AAS is a shorthand notation for the three Euler angles transforming the two frames into each other, here from the PAS to the AAS. It should not be confused with ΩLIS, introduced in Equation (3.36), which denotes the set of two polar angles specifying the orientation of an axis (IS) in a given frame (L). To proceed, we need to convert the chemical shielding Hamiltonian in Equation (5.22) to the general form discussed in the previous chapter that was given by Equation (4.37a). To do this, we need to express the spin and field factors as irreducible spherical tensors. This can be done formally by introducing a spin-field tensor, Xˆ , with Cartesian components Xˆ αβ = Iˆα Bβ (where a and b denote the Cartesian axes x, y, and z). The irreducible spherical tensor form of the spin-field tensor operators in the arbitrary axis system (AAS) becomes: Xˆ 2AAS ,0 =

1 6

(2 Bz Iˆz − Bx Iˆx − By Iˆy )

(5.26a)

ˆ ˆ ˆ ˆ  1  Xˆ 2AAS , ±1 = ∓ 2 Bz I x + Bx I z ± i(Bz I y + By I z )  

(5.26b)

ˆ ˆ ˆ ˆ  1  Xˆ 2AAS , ±2 = 2 Bx I x − By I y ± i(By I x + Bx I y )  

(5.26c)

If we choose to express the spin-field tensor in the laboratory frame, where BxLAB = ByLAB = 0 and BzLAB = B0 , the preceding expressions simplify to: Xˆ 2LAB ,0 =

Iˆ B

2 3 z 0

(5.27a)

1 ˆ Xˆ 2LAB , ±1 = ∓ 2 I ± B0

(5.27b)

Xˆ 2LAB , ±2 = 0

(5.27c)

We will incorporate the field strength, B0, into the interaction strength, xCSA, and define the operators, Tˆl ,− q , of Equation (4.37a) for the case of the CSA mechanism as: Tˆ2CSA ,0 =

8 3 z

Iˆ

(5.28a)

ˆ Tˆ2CSA , ±1 = ∓ I ±

(5.28b)

Tˆ2CSA , ±2 = 0

(5.28c)

In order to make a contraction of the rank-2 shielding tensor and rank-2 spin-field tensor, we must express both tensors in a common frame. We choose the laboratory frame for this purpose because this is the frame of

Applications of Redfield Theory to Systems of Spin 1/2 Nuclei

111

preference for the dominant Zeeman interaction. To rotate the shielding tensor into that frame, we use Equation (5.25) and note explicitly that the angle Ω PAS,LAB ≡ Ω I,L rotating the PAS of the shielding tensor into the laboratory frame is time dependent because of molecular tumbling: 2

σ 2LAB ,q =

∑D

2 p ,q

(Ω I ,L (t))σ 2PAS ,p

(5.29)

p=−2

which results in 2

Hˆ 1 (t) = ξCSA ′

2

∑ ∑ (−1) D q

2 p ,q

Sˆ (Ω I ,L (t))σ 2PA , p Tl ,− q

(5.30)

q=−2 p=−2

with the Tˆ -operators given in Equation (5.28a), Equation (5.28b), and Equation (5.28c) and the s PAS components by Equation (5.24a), Equation (5.24b), and Equation (5.24c). The symbol x′CSA is a constant related to units and normalization. We will soon replace it by another interaction strength parameter, defined later. The situation simplifies significantly if the shielding tensor is axially symPAS PAS = σ yy metric, σ xx or h = 0. The only nonzero element of s PAS is then the p = 0 term and the summation over p reduces to a single term. Under these conditions, the only Wigner rotation matrix elements that need to be considered in Equation (5.30) are D02,q and these are proportional to the spherical harmonics Y2,q. Equation (5.30) thus simplifies to: Hˆ 1 (t) = ξCSA

2

∑ (−1) Y q

2 ,q

(Ω LI (t))Tˆl ,− q

(5.31)

q=−2

With the spherical harmonics normalized as in Equation (4.44a), Equation (4.44b), and Equation (4.44c) and with the operators Tˆ given in Equations (5.28), the CSA interaction strength parameter to be used in Equation (5.31) becomes:

ξCSA = −(2π/15)1/2 γ I B0 ∆σ = (2π/15)1/2 ω I ∆σ

(5.32)

In general, the chemical shielding tensor is not axially symmetric. However, as discussed by Goldman,14 an arbitrary shielding tensor can be decomposed into a sum of two axially symmetric tensors. We have thus the choice of using the more complicated form of the CSA Hamiltonian given in Equation (5.30) or of working in terms of two axially symmetric shielding interactions on each nucleus. The Hamiltonian will then consist of two simpler terms, such as that of Equation (5.31), and the calculations of the relaxation behavior will require including the cross-correlation between the two CSA terms, in full analogy to the multiple dipolar interactions treated in Section 5.1.

112 5.2.3

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications The CSA Relaxation

Let us assume for simplicity that we deal with a single nucleus, characterized by an axial CSA. The unperturbed Zeeman Hamiltonian has the eigenstates |1〉 =| 1 2 , 1 2 〉 and |2 〉 =| 1 2 , − 1 2 〉 . We use this |I,m〉 notation to avoid confusing the common Greek letter symbols for these states with the indices in Equation (4.28) through Equation (4.36) and in their following analogues. The density matrix for such a simple system has four elements. Two of them are populations, r11 and r22, and two are coherences, r12 and r21. With this simple system, we can easily exemplify the use of Redfield theory in detail. The evolution of the four density matrix elements is in principle described by a 4 × 4 relaxation supermatrix, according to the formalism of Section 4.1. The situation simplifies, however, because of the secular approximation. The secular approximation removes couplings between populations and coherences, as well as between coherences with coherence orders +1 and –1. The full set of equations corresponding to Equations (4.30) therefore takes the following form:  ρ 11   R1111    d  ρ 22   R2211 = dt  ρ 12   0     ρ   0 21

R1122

0

R2222 0 0

0 R1212 0

0   ρ11    0   ρ22  0   ρ12  R2121   ρ21 

(5.33)

where we recognize the simplest possible form of the “Redfield kite” (compare Figure 4.1). We compute the elements of the relaxation supermatrix using Equation (4.28), which yields R1111 =

1 2

(J1111 (0) + J1111 (0) − J1111 (0) − J2121 (ω 0 ) − J1111 (0) − J2121 (ω 0 ))

= −J2121 (ω 0 )

(5.34a)

R2222 = −J1212 (ω 0 )

(5.34b)

R1122 = R1122 = J1212 (ω 0 )

(5.34c)

R1212 =

1 2

( 2J1122 (0) − J1111 (0) − J2222 (0) − J2121 (ω 0 ) − J1212 (ω 0 ))

(5.34d)

R2121 =

1 2

( 2 J2211 (0) − J1111 (0) − J2222 (0) − J2121 (ω 0 ) − J1212 (ω 0 ))

(5.34e)

Applications of Redfield Theory to Systems of Spin 1/2 Nuclei

113

In the first line of Equation (5.34a), we explicitly list all the six terms resulting from Equation (4.28), where both g = 1 and g = 2 contribute. The spectral densities Jaa′bb′(w) are, in turn, calculated using Equation (4.29). With the Hamiltonian of Equation (5.31), Equation (4.29) can be rewritten as: ∞

2

2 Jαα ′ ββ ′ (ω ) = 2ξCSA

∑ ∫ (−1)

q

α Y2 ,q (t)Tˆ2 ,− q α ′ β ′ Y2*,q (t + τ )Tˆ2†,− q β exp(−iωτ )dτ

q=−2 0

∞

2

2 = ξ CSA

∑

∫

(−1)q α Tˆ2 ,− q α ′ β ′ Tˆ2†,− q β 2 Y2 ,q (t)Y2*,q (t + τ ) exp(−iωτ )dτ

q=−2

0

2

∑ (−1)

2 = ξ CSA J l (ω )

q

α Tˆ2 ,− q α ′ β ′ Tˆ2†,− q β

q=−2

(5.35) The overbar indicates ensemble averaging. The spectral densities Jl(w) are strictly analogous to those occurring in the theory of the dipole–dipole relaxation; the only difference is that the axis whose reorientation we follow is the principal axis of the CSA rather than the dipole–dipole axis. To complete the derivation of the relaxation supermatrix, we need the matrix elements of the operators Tˆl ,− q , which can be obtained using the definitions of the step operators (Equation 3.12) and the Pauli spin matrices (Equations (1.4)). We collect the results in Table 5.1. Note that the operators Tˆ2 , ± 2 and their matrix elements vanish. Using Equation (5.35), we obtain for the longitudinal (population-related) part of the relaxation matrix: 2 R1111 = R2222 = −ξCSA J 2 (ω 0 )

(5.36a)

2 R2211 = R1122 = ξCSA J 2 (ω 0 )

(5.36b)

TABLE 5.1 Matrix Elements, 〈α |Tˆl ,− q |β 〉 , of the Operators Tˆl ,− q in Equations (5.28) 



〈T2,–1〉

1

1

0

1 2

2 1

0 1

2

2

0

〈T2,0〉

+

2 3

0 0

−

〈T2,1〉 0 –1 0

2 3

0

114

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

With this, the longitudinal part of the Redfield expression becomes: 2 2 d  ρ11   −ξ CSA J 2 (ω 0 ) ξ CSA J 2 (ω 0 )   ρ11  =      2 2 dt  ρ22   ξ CSA J 2 (ω 0 ) −ξ CSA J 2 (ω 0 )  ρ22 

(5.37)

As we did at the beginning of Section 2.1, we can choose to work in terms of a population difference (proportional to the expectation value of Iˆz ) and the sum of populations, which is constant in time. We obtain then: d 2 ( ρ11 − ρ22 ) = −2ξCSA J 2 (ω 0 )( ρ11 − ρ22 ) dt

(5.38a)

d ˆ 2 I z = −2ξCSA J 2 (ω 0 ) Iˆz dt

(5.38b)

or

After correcting for the finite temperature (compare Section 3.5), Equation (5.38b) becomes analogous to the longitudinal Bloch equation, Equation (1.22a), with: 1 2 = 2ξCSA J 2 (ω 0 ) = 4π (γ I B0 )2 ( ∆σ )2 J 2 (ω 0 ) 15 T1,CSA

(5.39)

τc = 2 (γ I B0 )2 ( ∆σ )2 15 1 + ω 02τ c2

In the second equality, we have assumed that the molecule of interest tumbles as a rigid spherical object, with the correlation time tc, and used the normalized spectral density of the form J 2 (ω 0 ) = (2π )−1 τ c/(1 + ω 02τ c2 ) . We will return to the derivation of model-dependent forms of the spectral density function in Chapter 6. The specific form of J2(w0) in Equation (5.39) will be derived in Section 6.1. In the same way, we can calculate the decay rate constant for the two coherences, R1212 = R2121 (the reader is encouraged to do that calculation), and identify them with the transverse relaxation rate: 1 2 = ξCSA T2 ,CSA

(

4 3

)

J 2 (0) + J 2 (ω 0 ) = 2π (γ I B0 )2 ( ∆σ )2 15

 τc  = 1 (γ I B0 )2 ( ∆σ )2  4 τ c +  15 1 + ω 02τ c2  3

(

4 3

J 2 (0) + J 2 (ω 0 )

) (5.40)

Applications of Redfield Theory to Systems of Spin 1/2 Nuclei

115

The effects of a nonaxial CSA tensor are easy to include in the simple case of isotropic reorientation: the rate expressions in the second lines of Equation (5.39) and Equation (5.40) are only multiplied by the factor (1 + h2/3). Some interesting observations can be made concerning the relaxation rates of the CSA mechanism as given by Equation (5.39) and Equation (5.40). First, we note that the rates are proportional to the square of the external magnetic field. In the extreme narrowing range, this field dependence can be used for separating the contributions from the CSA and DD relaxation mechanisms. By measuring the relaxation rate at several magnetic field strengths and plotting the measured rate against the square of the magnetic field, we obtain a straight line with the abscissa equal to the dipolar rate. Outside the extreme narrowing range, this B02-dependence will be offset by the term ω 02τ c2 in the denominators of Equation (5.39) and Equation (5.40). In the limit of ω 02τ c2 >>1, the longitudinal rate will in fact become independent of the magnetic field. Apart from the magnetic field, the CSA relaxation depends on the magnitude of the shielding anisotropy. Large chemical shielding anisotropies are characteristic for many heteronuclei. In the case of 13C, large ∆s values (on the order of 200 ppm) are characteristic for carbons participating in multiple bonds — for example, in carbonyl groups or in aromatic rings. Relaxation of such nuclei at high magnetic fields of the present day NMR spectrometers can be dominated by the CSA mechanism, in particular for carbon nuclei that are not directly bound to protons. Another nuclear species for which the CSA relaxation is important is nitrogen-15 in amide groups in peptides and proteins. Another interesting property of the CSA relaxation is that the longitudinal and transverse relaxation rates are not equal to each other, even in extreme narrowing, where the ratio T1/T2 = 7/6. This is in contrast to the dipole–dipole relaxation and most other relaxation mechanisms and is the result of the properties of the CSA Hamiltonian under spatial rotation. We note, however, that at the limit of zero magnetic field, where the three Cartesian directions become equivalent in isotropic space, both relaxation rates vanish.

5.2.4

The DD/CSA Interference

Even though the elements T2, ± 2 are zero in the case of CSA, because of the nature of the spin-field tensor, it is important to keep in mind that the Hamiltonian in Equation (5.31) is a scalar contraction of rank-2 tensors. An important consequence of this fact is that the CSA Hamiltonian and the DD Hamiltonian of Equation (3.13) have the same properties under rotation of coordinate systems and both depend on the spherical harmonics of the same rank. Because of this property, the CSA and dipolar interaction can interfere with each other, in full analogy with the case of multiple dipolar interactions discussed in Section 5.1. These types of cross-correlation effects can in the NMR context be traced back to the work of Shimizu in the early 1960s15 and have been discussed in the review by Kumar et al.1 Consider a system consisting

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Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

of two nuclear spins, I and S, with the spin quantum number 1/2. We treat the two Zeeman interactions and the scalar spin–spin coupling as the unperturbed Hamiltonian, while the dipolar interaction and the two CSA interactions enter the perturbation Hamiltonian: 2

2

Hˆ 1 (t) = ξISDD

∑

(−1)q Y2 ,q (Ω LIS (t))Tˆ2IS,− q + ξICSA

q=−2

∑ (−1) Y q

2 ,q

(Ω LI (t))Tˆ2I,− q

q=−2

2

+ ξSCSA

∑ (−1) Y q

2 ,q

(5.41)

(Ω SL (t))Tˆ2S,− q

q=−2

The interaction strength constants are defined in full analogy with Equation (3.13) and Equation (5.31), including the definitions of the irreducible tensor operators of Equations (4.43) and Equations (5.28). The set of angles Ω LIS (t) describes the orientation of the IS axis with respect to the laboratory frame; the angles Ω LI (t) and Ω SL (t) refer to the orientation of the principal axis of the CSA for the I- and S-spins, respectively, in the same frame. Both shielding tensors are assumed to be axially symmetric. It is useful to transform the orientational part of the three terms in Equation (5.41) in such a way that they all contain a spherical harmonic of the same time-dependent angle. We will come back to the calculations of spectral densities, starting with this more complicated form of Hamiltonian involving several coordinate frame transformations in Chapter 6. The result of applying the Redfield theory to the longitudinal part for this four-level system, analogous to that of Figure 3.5, is a relaxation equation, most conveniently expressed in the operator or magnetization mode formalism as:  < Iˆz >   Γ 11  d  ˆ   <Sz > = −  Γ 21 dt    Γ 31  < 2 Iˆz Sˆ z >

Γ 12 Γ 22 Γ 32

eq   ˆ Γ 13   < I z > − I z   Γ 23   <Sˆ z > −Szeq    Γ 33   < 2 Iˆz Sˆ z > 

(5.42)

The elements Γ11 and Γ22 of the relaxation matrix are sums of the corresponding elements occurring in the Solomon equations, Equation (3.20a) and Equation (3.20b) (the dipolar rI and rS), and the CSA spin-lattice relaxation rates as given by Equation (5.39). Γ12 and Γ21 are identical to the dipolar cross-relaxation rate, sIS; the CSA does not contribute to these elements. Γ33 is the decay rate of the longitudinal two-spin order, 〈2 IˆzSˆ z 〉 . The elements connecting the expectation values of one-spin operators 〈 Iˆz 〉 and 〈Sˆ z 〉 with the two-spin operator 〈2 IˆzSˆ z 〉 — Γ13, Γ23, Γ31, and Γ32 — are the result of interference or cross-correlation between the dipolar interaction and one of the CSA interactions. It is interesting to note that these terms couple the expectation values of operators with opposite symmetries under spin inversion, contrary to the

Applications of Redfield Theory to Systems of Spin 1/2 Nuclei

117

case of interferences between dipolar interactions, which couple expectation values of operators of the same symmetry. These terms arise through interference between the DD and CSA Hamiltonians. These CCRRs can be derived using the formulation of Section 4.1, a detailed derivation can be found in the review by Smith et al.16, or using the operator approach of Section 4.2. The latter derivation can be found in the paper by Goldman.14 The results are: Γ 13 = Γ 31 = δ I , IS = ξISDDξICSA J IS, I (ω I )

(5.43a)

Γ 23 = Γ 32 = δ S, IS = ξISDDξSCSA J IS,S (ω S )

(5.43b)

For the sake of completeness, we also provide an expression for Γ33: Γ 33 = ρIS =

1 8

(ξ )

DD 2 IS

(

J IS, IS (ω I ) +

)

CSA 2 I

+2 ξ

1 8

(ξ )

DD 2 IS

(

)

J IS, IS (ω S )

CSA 2 S

J I , I (ω I ) + 2 ξ

(5.43c) J S,S (ω S )

JIS,IS(w) are autocorrelated dipolar spectral densities, identical to those used in Chapter 3 and in Section 5.1; JI,I(w) and JS,S(w) are autocorrelated CSA spectral densities analogous to those appearing in Equation (5.39). For a rigid, isotropic rotor with a rotational correlation time tc, the cross-correlated spectral densities take the form:

(

)

(5.44a)

(

)

(5.44b)

2τ c J IS, I (ω ) = 1 3 cos 2 θ IS, I − 1 1 2 4π 1 + ω 2τ c2 2τ c J IS,S (ω ) = 1 3 cos 2 θ IS,S − 1 1 4π 1 + ω 2τ c2 2

Notice that we use here the same normalization of the spectral density as in Equation (3.16), i.e., we include the factor 1/4p. The reader should be aware that different normalizations are used in the literature; the one omitting the 1/4p is rather popular. The factors 21 (3 cos 2 θ IS, I − 1) and 1 (3 cos 2 θ IS,S − 1) occur as a result of the time-independent coordinate trans2 formations between the principal frames of the DD and CSA (for the I- and S-spins, respectively) interactions. The angles qIS,I and qIS,S are between the principal axes of the two CSA tensors and the DD internuclear vector. Replacing the symbols x as strength constants with the dipolar coupling constant and shielding anisotropy and setting in the spectral densities according to

118

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

Equation (5.44a) and Equation (5.44b), the CCRR of Equation (5.43a) becomes:

)(

)

τc δ I , IS = − 1 bIS ω I ∆σ 3 cos 2 θ IS, I − 1 5 1 + ω I2τ c2

(

(5.45)

with an expression for dIS,S obtained mutatis mutandis. An interesting feature of Equation (5.43a), Equation (5.43b), and Equation (5.43c) is that the CCRRs contain products of the interaction strength constants, in analogy with Equation (5.15). The consequence of this fact is that a strong dipolar interaction can help to make a weak CSA visible. In other words, the DD/CSA CCRRs can be used to estimate the shielding anisotropy, ∆s, in solution, even if the anisotropy is so small that the contribution of the CSA mechanism to the spin-lattice relaxation rate (proportional to the square of ∆s) is negligible. As an example, the chemical shift anisotropy for the carbon and proton in chloroform, ∆sH and ∆sC, was determined through measurements of longitudinal DD/CSA CCRRs17 to be about 12 ppm and 32 ppm, respectively. At the magnetic field of 9.4 T, this corresponds to |ξHCSA | = 20 · 103 rad s–1 (3.1 kHz) DD and |ξCCSA | = 13 · 103 rad s–1 (2.1 kHz), compared to |ξCH | = 51 · 104 rad s–1 (81 kHz). It is also worth noting that the proton CSA interaction strength constant is larger than its carbon counterpart, even though the ∆s values expressed in parts per million show the opposite relation. This is an effect of the larger proton magnetogyric ratio. In analogy with the case of dipolar interference effects, the DD/CSA CCRRs give rise to asymmetries in multiplet relaxation. We will return to this point in the context of experimental methods.

5.2.5

Transverse Relaxation Effects of the DD/CSA Interference

The DD/CSA interference effects also influence the transverse relaxation and the spectral lineshapes in the sense that they contribute to unequal line broadening of the multiplet components. For a detailed discussion, we refer to the paper by Goldman.14 Here, we simply wish to summarize certain features of the results. The J-coupled spectrum can in this case be described very similarly to the case discussed at the end of Section 4.2. The I-spin spectrum is characterized in terms of two coherences, which we can express in several ways. One convenient possibility is to work with expectation values of 〈 Iˆ+ 〉 and 〈2 Iˆ+Sˆ z 〉 . The first of these expectation values corresponds to an in-phase doublet, split by the J-coupling to the S-spin and the second is an antiphase doublet. Alternatively, we can formulate linear combinations of these operators corresponding to individual transitions of the doublet. The evolution of the two expectation values follows a set of two coupled equations, reminding us of Equation (4.64a) and Equation (4.64b), with the coupling terms

Applications of Redfield Theory to Systems of Spin 1/2 Nuclei

119

containing the decay rates, the transverse cross-relaxation rates, and the transverse cross-correlated relaxation rates. If the doublet components are well separated, the cross-relaxation effects can be neglected. Both lines are then expected to have Lorentzian lineshapes. In the absence of the CSA, the two doublet components have the same width. The DD/CSA cross-correlation has a peculiar effect on the two linewidths: it adds another term, h, to the linewidth of one of the doublet components, while the same term is subtracted from the linewidth of the second component. This phenomenon is referred to as differential line broadening (DLB). The difference in the linewidth, expressed in radians per second, is given by: 2η = ξISDDξICSA  J IS, I (ω I ) + 34 J IS, I (0) 

(5.46)

The DLB, a characteristic signature of the DD/CSA interference, is illustrated in Figure 5.3. If we carefully inspect Equation (5.41), we can conclude the IS-spin system might be influenced, at least in principle, by still another interference effect: that between the two CSA interactions. This interference effect does not influence the longitudinal relaxation or the transverse relaxation of single quantum coherences. In a scalar-coupled IS two-spin system, however, one can also excite and measure the relaxation of double-quantum (DQ) and zero-quantum (ZQ) coherences. (We will discuss how this can be done experimentally in Chapter 10.) Each of these coherences is characterized by a decay rate constant, R2DQ and R2ZQ, related to the corresponding linewidth. The general properties of these multiple-quantum (MQ) coherence linewidths, as contrasted against the directly observable single-quantum coherences, were discussed in an important paper by Wokaun and Ernst.18 The MQ relaxation rates are affected by the autocorrelated dipolar and CSA spectral densities, but not by the DD/CSA interference. (An elegant symmetry argument for that was given by Werbelow.19) On the other hand,

FIGURE 5.3 Differential line broadening.

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Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

as indicated in the same article by Werbelow and demonstrated experimentally by Pellecchia and coworkers,20 the DQ and ZQ coherences are affected by the CSA/CSA interference. In fact, the magnitude of this interference effect can be obtained by comparing the linewidth of these two transitions. The measured effects are not strongly pronounced, however, because the coupling is between two interactions with relatively weak interaction strengths.

5.2.6

The Antisymmetric Component

We now return to the rank 1, antisymmetric component of the shielding tensor. The relaxation effects of this component were proposed for the first time by Blicharski21 and were investigated thoroughly by Anet and O’Leary, who also described them in a review.12,13 See that work for details; we choose to present only the final results here. The antisymmetric shielding contributes to relaxation in a rather simple way, adding another term to the total spin-lattice and spin–spin relaxation rates. The contributions have the form: 1 anti 1,CSA

T

(

 = 1 γ 2 B02  σ xy − σ yx 6 

) + (σ 2

xz

− σ zx

) + (σ 2

yz

2 τ1 − σ zy   1 + ω 02τ 12

)

(5.47a)

and 1 anti 2 ,CSA

T

(

 = 1 γ 2 B02  σ xy − σ yx 12 

) + (σ 2

xz

− σ zx

) + (σ 2

yz

2 τ1 − σ zy   1 + ω 02τ 12

)

(5.47b)

The correlation time, t1, is not the same as the tc that we encountered in the case of the DD interaction or the symmetric CSA. Here, it is the characteristic correlation time for rank-1 spherical harmonics, mentioned in Section 2.4. As will be shown in Section 6.1, for isotropic rotational diffusion, we have the relation t1 = 3tc. Two interesting observations can be made concerning the transverse relaxation rate contribution: first, in analogy with the symmetric CSA, it is not equal to the corresponding contribution to the longitudinal relaxation rate. Second, it has an unusual property for transverse relaxation because it does not depend on the zero-frequency spectral density. Kowalewski and Werbelow22 estimated the possible effect of the antisymmetric shielding anisotropy relaxation in peptides and proteins, using the shielding tensor from ab initio calculations on formamide. The contributions of Equation (5.47a) and Equation (5.47b) were found to be negligible for the case of isotropic motion and moderate deviations from it. The relative importance of the antisymmetric shielding contributions might, however, increase dramatically in cases of strongly anisotropic motions (compare Chapter 6).

Applications of Redfield Theory to Systems of Spin 1/2 Nuclei

5.3

121

Spin-Rotation Relaxation

The next source of relaxation to deal with is the spin-rotation interaction. This relaxation mechanism is most important for spins in small symmetric molecules. The theory of relaxation through this interaction has been reviewed by McClung.23 Detailed derivations of the spin-rotational relaxation can also be found in the paper by Hubbard24 and in the book by McConnell (see further reading). The origin of the spin-rotation interaction is the fact that a rotating electronic charge distribution produces a magnetic field. The size of this magnetic field is proportional to the magnitude of the rotational angular momentum. The rotationally induced magnetic field can interact with the nuclear spins. The spin-rotation Hamiltonian can be expressed as: Hˆ ISR = Iˆ ⋅ C ⋅ Jˆ

(5.48)

where Iˆ is the nuclear spin operator (a vector or a rank-1 spherical tensor) ˆJ is the angular momentum operator related to the rotation of the molecule (another vector or a rank-1 tensorial quantity) C is the rank-2 spin-rotation coupling tensor. We assume, as elsewhere in this book, that the angular momentum operators are dimensionless and that the elements of the spin-rotation coupling tensor have the units rad s–1. The Cartesian element, Cxy , is proportional to the magnitude of the magnetic field in the x direction caused by the rotation around the molecular y-axis. In analogy with the case of the CSA interaction, we must be careful with coordinate frames. The spin-rotation coupling tensor is most conveniently expressed in its principal axis system, fixed in the molecule. We assume for simplicity that the tensor is axially symmetric; that is, it can in its principal frame be fully characterized by two components, one parallel to the axis, C||, and another perpendicular to the axis, C⊥. The molecular frame is also appropriate for the rotational angular momentum. We therefore transform the spin operators, in irreducible spherical form (see Equation 4.41a and Equation 4.41b), into the molecular frame: 1

Iˆkmol =

∑ Iˆ

1, q

Dq1, k (Ω L ,mol )

(5.49)

q=−1

The Euler angles Ω L,mol describe the transformation of the laboratory frame into the molecular frame. In analogy with the cases discussed earlier, the angles ΩL,mol are time dependent through molecular tumbling. This is, however,

122

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

not the only source of random time dependence in Equation (5.48). The angular momentum, related to molecular rotation in condensed matter, also changes randomly. A common assumption is that the rotational motion of the molecule in a liquid can be treated classically and that the operator Jˆm (the m-th component of the operator ˆJ introduced in Equation 5.48) at time t can be replaced by Imwm(t), where Im is the m-th principal component of the moment of inertia tensor and wm(t) is the corresponding angular velocity, with the stochastic time dependence now residing in the angular velocity. In a detailed analysis, one must keep in mind that the principal axis system of the moment of inertia tensor and that of the spin-rotation tensor do not necessarily coincide. Reformulating the spin-rotation Hamiltonian in the form of Equation (4.37a) is tedious and the reader is referred to the paper by Hubbard24 or the book by McConnell for the full derivation. Here, it suffices to state that it is possible and that the resulting classical function Vl,q(t) is time dependent through the orientation of the molecular frame, with respect to the external magnetic field, and the angular velocity. The calculation of the time-correlation functions and spectral densities corresponding to these complicated quantities is also tedious. Assuming that the orientation of the molecule at any instant and its angular velocity are uncorrelated, Hubbard obtained that the total time-correlation function can be expressed in terms of an angular velocity correlation function, 〈wm(t)wm(t + t)〉, and correlation functions for reorientation of rank-2 spherical harmonics of the type encountered before, in the context of the DD and CSA relaxation. Assuming that the molecule is a spherical top with the moment of inertia, I, and that the molecular motion is isotropic rotational diffusion in small steps, the angular velocity correlation function becomes a decaying exponential with the time constant (correlation time) tJ. This is simply related to the rotational correlation time for rank-2 spherical harmonics, tc, through what is known as the Hubbard relation:

τ Jτ c =

I 6 kBT

(5.50)

The angular velocity (or angular momentum) correlation time is usually much shorter than the rotational correlation time. For the cases in which spinrotation relaxation may be important, for small symmetric molecules, both correlation times are short enough to guarantee the conditions of extreme narrowing. We obtain then that the spin–rotation interaction gives rise to an exponential relaxation of the longitudinal and the transverse components of the nuclear magnetization, following the Bloch equations, with:  2 IkBT  1 1 = = 2C⊥2 + C2 τ J T1,SR T2 ,SR  3 2 

(

)

(5.51)

Applications of Redfield Theory to Systems of Spin 1/2 Nuclei

123

We will return to the spin-rotation relaxation in our discussion of relaxation in gas phase in Chapter 16.

5.4

Other Relaxation Mechanisms for Spin 1/2 Nuclei

In addition to the previously mentioned anisotropic interactions, which vary with time through processes related to molecular rotation, one more type of random modulation needs to be considered in the context of nuclear spin relaxation: chemical exchange. Chemical exchange means that the nuclear spins change their environment through a chemical process, which can be intramolecular (conformational exchange) or intermolecular. The exchange processes modulate anisotropic interactions of the type mentioned earlier, as well as isotropic interactions such as chemical shift and J-coupling. Some of the exchange effects, which are important in the first place for transverse relaxation, can be described using Redfield theory, but others need to be dealt with in other ways. We choose to discuss all of these various aspects of chemical exchange in Chapter 13. In the case of dipolar or scalar IS coupling, a rapid relaxation of the S-spin through some other independent and efficient mechanism provides also a possible modulation of the two-spin interaction, which can contribute to the I-spin relaxation. We will return to these effects in Chapter 14 and Chapter 15, discussing the quadrupolar nuclei and paramagnetic systems. In the case of indirect spin–spin coupling modulated by chemical exchange or S-spin relaxation, we usually speak of scalar relaxation of the first or second kind in the terminology of Abragam (see further reading). The indirect spin–spin coupling interaction can contribute to relaxation in one more way. The interaction is, in fact, more complicated than indicated by Equation (4.66). Besides the term J IS Iˆz Sˆ z (and the nonsecular terms J IS IˆxSˆ x and J IS IˆySˆ y) corresponding to the scalar product of the two spin operators, we should take into account that the indirect coupling can in principle have an anisotropic, rank-2 tensorial component similar to the dipolar coupling tensor. Actually, according to some authors (for example, Levitt; see further reading), the indirect spin–spin coupling is referred to as indirect dipole–dipole interaction. In the corresponding Hamiltonian, analogous to the dipolar Hamiltonian of Equation (3.13), the J-tensor anisotropy expressed in rad s–1, 2p∆J, takes the place of the dipole-coupling constant. The anisotropy of the indirect coupling tensor is a very small correction to the direct dipolar coupling for light spin 1/2 nuclei, but the situation may turn out to be different for heavy nuclei, in which the isotropic J-couplings may be very large and their anisotropies unknown. Summarizing the chapter on relaxation mechanisms for spin 1/2 nuclei, we conclude that the simple additivity of contributions from different relaxation

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Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

mechanisms to the total observed spin-lattice relaxation rate often found in NMR books: 1 1 1 1 1 1 = + + + + T1,obs T1,DD T1,CSA T1anti T T 1,SR 1,SC ,CSA

(5.52)

where 1/T1,DD may include the J-coupling anisotropy and 1/T1,SC refers to the scalar relaxation, is only valid under certain conditions. The conditions are that the cross-relaxation effects are suppressed through the experimental set-up (which usually is the case, for example, for carbon-13 and nitrogen-15 relaxation under broadband decoupling of protons) and that the cross-correlation effects are absent or suppressed. A similar statement is true for the transverse relaxation, in which additional caution should be exercised in the presence of chemical exchange. Under more general conditions, one should work in terms of magnetization modes (or a similar approach) and evaluate the contributions to different elements of the relaxation matrix, following the statement from the important paper by Szymanski et al.25 that the concept of a relaxation mechanism should refer to a pair of (identical or different) interactions rather than to a single one.

References 1. Kumar, A., Grace, R.C.R., and Madhu, P.K. Cross-correlations in NMR, Prog. Nucl. Magn. Reson. Spectr., 37, 191–319, 2000. 2. Werbelow, L.G. Relaxation processes: cross correlation and interference terms, in Encyclopedia of Nuclear Magnetic Resonance, Grant, D.M. and Harris, R.K., Eds. (Chichester: Wiley), 4072–4078, 1996. 3. Brutscher, B. Principles and applications of cross-correlated relaxation in biomolecules, Concept Magn. Reson., 12, 207–229, 2000. 4. Werbelow, L.G. and Grant, D.M. Intramolecular dipolar relaxation in multispin system, Adv. Magn. Reson., 9, 189–299, 1977. 5. Canet, D. Relaxation mechanisms: magnetization modes, in Encyclopedia of Nuclear Magnetic Resonance, Grant, D.M. and Harris, R.K., Eds. (Chichester: Wiley), 4046–4053, 1996. 6. Hubbard, P.S. Nuclear magnetic relaxation of three and four spin molecules in a liquid, Phys. Rev., 109, 1153–1158, 1958. 7. Brondeau, J. and Canet, D. Longitudinal magnetic relaxation of carbon-13 (or nitrogen-15) interacting with a strongly irradiated proton system, J. Chem. Phys., 67, 3650–3654, 1977. 8. Kowalewski, J., Effemey, M., and Jokisaari, J. Dipole–dipole coupling constant for a directly bonded CH pair — a carbon-13 relaxation study, J. Magn. Reson., 157, 171–177, 2002. 9. Smith, S.A., Palke, W.E., and Gerig, J.T. The Hamiltonians of NMR. Part I, Concepts Magn. Reson., 4, 107–144, 1992.

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125

10. Smith, S.A., Palke, W.E., and Gerig, J.T. The Hamiltonians of NMR. Part II, Concepts Magn. Reson., 4, 181–204, 1992. 11. Smith, S.A., Palke, W.E., and Gerig, J.T. The Hamiltonians of NMR. Part III, Concepts Magn. Reson., 5, 151–177, 1993. 12. Anet, F.A.L. and O’Leary, D.J. The shielding tensor. Part I: Understanding its symmetry properties, Concepts Magn. Reson., 3, 193–214, 1991. 13. Anet, F.A.L. and O’Leary, D.J. The shielding tensor. Part II. Understanding its strange effects on relaxation, Concepts Magn. Reson., 4, 35–52, 1992. 14. Goldman, M. Interference effects in the relaxation of a pair of unlike spin-1/2 nuclei, J. Magn. Reson., 60, 437–452, 1984. 15. Shimizu, H. Theory of the dependence of nuclear magnetic relaxation on the absolute sign of spin–spin coupling constant, J. Chem. Phys., 40, 3357–3364, 1964. 16. Smith, S.A., Palke, W.E., and Gerig, J.T. The Hamiltonians of NMR. Part IV: NMR relaxation, Concepts Magn. Reson., 6, 137–162, 1994. 17. Mäler, L. and Kowalewski, J. Cross-correlation effects in the longitudinal relaxation of heteronuclear spin systems, Chem. Phys. Lett., 192, 595–600, 1992. 18. Wokaun, A. and Ernst, R.R. The use of multiple quantum transitions for relaxation studies in coupled spin systems, Mol. Phys., 36, 317–341, 1978. 19. Werbelow, L.G. The use of interference terms to separate dipolar and chemical shift anisotropy contributions to nuclear spin relaxation, J. Magn. Reson., 71, 151–153, 1987. 20. Pellecchia, M., Pang, Y.X., Wang, L.C., Kurochkin, A.V., Kumar, A., and Zuiderweg, E.R.P. Quantitative measurement of cross-correlations between N-15 and (CO)- C-13 chemical shift anisotropy relaxation mechanisms by multiple quantum NMR, J. Am. Chem. Soc., 121, 9165–9170, 1999. 21. Blicharski, J.S. Nuclear magnetic relaxation by anisotropy of the chemical shift, Z. Naturforsch. A, 27, 1456–1458, 1972. 22. Kowalewski, J. and Werbelow, L. Evaluation of spin relaxation induced by chemical shielding anisotropy: a comment on the importance of the antisymmetric component, J. Magn. Reson., 128, 144–148, 1997. 23. McClung, R.E.D. Spin-rotation relaxation theory, in Encyclopedia of Nuclear Magnetic Resonance, Grant, D.M. and Harris, R.K., Eds. (Chichester: Wiley), 4530–4535, 1996. 24. Hubbard, P.S. Theory of nuclear magnetic relaxation by spin-rotational interactions in liquids, Phys. Rev., 131, 1155–1165, 1963. 25. Szymanski, S., Gryff–Keller, A.M., and Binsch, G. A Liouville space formulation of Wangsness–Bloch–Redfield theory of nuclear spin relaxation suitable for machine computation. I. Fundamental aspects, J. Magn. Reson., 68, 399–432, 1986.

6 Spectral Densities and Molecular Dynamics

CONTENTS 6.1 Time-Correlation Function for Isotropic Small-Step Rotational Diffusion ..................................................................................128 6.2 More General Rigid Objects.....................................................................134 6.2.1 The Asymmetric and Symmetric Top ........................................134 6.2.2 Physical Interpretation of Correlation Times ...........................140 6.3 Dynamic Models for Nonrigid Molecules.............................................142 6.3.1 General ............................................................................................142 6.3.2 Woessner’s Models .......................................................................144 6.3.3 The “Model-Free” Approach.......................................................149 References ............................................................................................. 155

Spectral densities are a central concept in NMR relaxation theory, as we saw in the beginning of this book (Chapter 2). They provide the tool through which measurements of relaxation parameters convey the information on molecular dynamics. Until now, we have introduced the spectral density function for the simple model describing motion of a rigid spherical object. In this chapter, we deal with the theory of time-correlation functions and spectral densities for different motional models, starting again with rigid, spherical objects and moving towards increasingly complex systems. Throughout this chapter, we limit ourselves to dealing with ordinary, isotropic liquids, i.e., systems whose macroscopic properties do not depend on orientation. (We will mention briefly the case of anisotropic media in Chapter 16.) In Section 6.1 and Section 6.2, we concentrate on rigid molecular systems characterized by reasonably rapid motions, and in Section 6.3 we introduce the spectral density functions for molecules undergoing internal motion. Experimental approaches to studies of molecular dynamics are presented in Chapter 11, in which the details of the physical information obtainable from spin relaxation are explored.

127

128

Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

6.1 Time-Correlation Function for Isotropic Small-Step Rotational Diffusion Several presentations on the subject of simple time-correlation functions exist in the literature — for example, the detailed description by Hennel and Klinowski (see the further reading section in the Preface). A less detailed but similar presentation is also given in the book by Abragam (see further reading). The concepts behind the derivation can be traced back to Debye’s classical work on dielectric relaxation1 and the paper by Bloembergen et al.2 We will start this chapter by dealing with a particularly simple system. The simplest possible case of reorientational motion in condensed matter is that of a rigid, spherical object undergoing a rotational diffusion process in small steps, as illustrated in Figure 2.1. In the theory of relaxation through rank-2 interactions, such as the dipole–dipole interaction and chemical shielding anisotropy (as well as the quadrupolar interaction, to be discussed in Chapter 14), a time-correlation function for the normalized spherical harmonics Yl,m with l = 2 is required. The case of m = 0 was introduced in Chapter 3 (compare Equation 3.14): Y2 ,0 =

1 4

5 π

(3 cos 2 θ − 1)

(6.1)

Here, θ is the angle between a molecule-fixed vector and the external magnetic field. To specify the direction of the vector in the laboratory frame fully, we also need a second angle, φ. We will use the symbol Ω (solid angle) for the full description of the direction of the vector, including both angles. The time-correlation function that we need to evaluate is: G2 (τ ) = 〈Y2 ,0 (t)Y2*,0 (t + τ )〉 =

∫ ∫Y

2 ,0

(Ω 0 )Y2*,0 (Ω)P(Ω 0 )P(Ω 0 |Ω , τ )dΩ 0dΩ

(6.2)

The subscript 2 on the time-correlation function refers to the fact that we are dealing with spherical harmonics with l = 2. As we will see later, the correlation function is not dependent on the value of m. The symbol Ω 0 stands for the orientation of the axis of interest at time zero and Ω at time τ. The complex conjugation sign on Y*2,0 is, strictly speaking, unnecessary because the function is real. We introduce it to make it easier to generalize the expression, for future use in more complicated cases. P(Ω 0) is the probability density of the initial orientation specified by the solid angle Ω 0; P(Ω 0|Ω ,τ) is the conditional probability density that the axis will be oriented at the angle Ω at time τ, provided that it was Ω 0 at time zero. We begin the calculation by looking at P(Ω 0). The problem of calculating this property becomes very simple if we assume that the medium in which the molecule reorients is isotropic. This assumption is fulfilled in all ordinary

Spectral Densities and Molecular Dynamics

129

liquids and is much less stringent than the assumption of isotropic motion. In isotropic medium, all angles Ω are equally probable: P(Ω ) = P(Ω 0) = 1/4π

(6.3a)

This particular value for the uniform probability is consistent with the condition that the integral over all space of a probability density function must be equal to unity:

∫

2π π

P(Ω 0 )dΩ 0 =

∫∫ 0 0

allspace

1 sin θ dθ dφ = 4π

π

∫ 2 sin θ dθ = 1 1

(6.3b)

0

Substituting Equation (6.3a) into Equation (6.2), we obtain: G2 (τ ) =

1 4π

∫∫Y

2 ,0

(Ω 0 )Y2*,0 (Ω)P(Ω 0 |Ω , τ )dΩ 0dΩ

(6.4)

Our next task is to find P(Ω 0 |Ω ,t), the conditional probability density or the propagator. To do this, we need to examine the motion of our system. The physical model with which we deal — small step rotational diffusion or diffusion on the surface of a sphere — is a special case of general three-dimensional diffusion. Fick’s second law of diffusion describes the three-dimensional diffusion process in isotropic medium:  ∂ 2 f ( x , y , z) ∂ 2 f ( x , y , z) ∂ 2 f ( x , y , z)  ∂f ( x , y , z) = D + +  ∂t ∂ z2 ∂ x2 ∂ y2  

(6.5a)

The function f is a measure of the concentration of diffusing particles at a certain point (x, y, z) and is a function of time. D is a diffusion coefficient. Equation (6.5a) can be transformed to spherical polar coordinates (r, θ, φ). This transformation is analogous to that performed in order to separate variables in the solution of the time-independent Schrödinger equation for the hydrogen atom (compare Atkins and Friedman in further reading). By concentrating on the angular part and by assuming that r is constant and equal to the radius of the sphere, we obtain Fick’s law for rotational diffusion:

∂ f (Ω , τ ) = DR ∆ˆ R f (Ω , τ ) ∂τ

(6.5b)

Now f is a measure of the concentration of particles at a certain solid angle Ω and DR is the rotational diffusion coefficient. ∆ˆ R is an expression in θ and φ, called Legendre operator (see Atkins and Friedman). The Legendre

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Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

operator is closely related to the total orbital angular momentum operator, Lˆ2. In analogy with the Lˆ2 operator, ∆ˆ R also has spherical harmonics as eigenfunctions, but with eigenvalues of the opposite sign, −l(l + 1): ∆ˆ RYl ,m = −l(l + 1)Yl ,m

(6.6)

In order to be useful as a propagator, f(Ω ,τ) must fulfill two additional conditions. It must be normalized in such a way that:

∫ f (Ω, τ )dΩ = 1

(6.7)

and the solid angle Ω at τ = 0 must be identical to Ω 0: f (Ω , 0) = δ (Ω − Ω 0 )

(6.8)

δ(x − a) is the Dirac delta function, with the following definition: ∞

∫ f (x)δ(x − a)dx = f (a)

(6.9)

−∞

We can say that when the δ function is multiplied with another function and the product is integrated, the operation “projects” out the value of that other function at the point where the argument of the δ-function is zero. The properties of the Dirac delta function are discussed in the book by Hennel and Klinowski (further reading); we will use the fact that the d-function can be expanded in any complete and orthonormal set of functions:

δ ( x − x ′) =

∑ ϕ (x)ϕ (x′) n

* n

(6.10a)

n

The spherical harmonics form a complete and orthonormal set of functions of the solid angle:

∫Y

∗ l ,m

(Ω)Yl′ ,m′ (Ω)dΩ = δ ll′δ mm′

(6.10b)

This allows us to expand the delta function of Equation (6.8) as:

δ (Ω − Ω 0 ) =

∑Y

l ,m

l ,m

(Ω)Yl*,m (Ω 0 )

(6.10c)

Spectral Densities and Molecular Dynamics

131

With the conditions specified by Equation (6.8) and Equation (6.9), the solution to the rotational diffusion equation becomes acceptable as the propagator P(Ω 0|Ω ,t). To find this propagator, we will use the fact that the spherical harmonics form a complete set of eigenfunctions to ∆ˆ R . We can expand f(Ω ,τ) at a certain time t ′ in spherical harmonics, which gives: ∞

f (Ω , τ ′) =

l

∑∑B

Y (Ω)

l ,m l ,m

(6.11)

l = 0 m=− l

For other τ values, the coefficients Bl,m are different and Yl,m are the same (time independent). In general, we have: ∞

f (Ω , τ ) =

l

∑∑B

l ,m

(τ )Yl ,m (Ω)

(6.12)

l = 0 m=− l

and we only need to compute the time-dependent Bl,m(τ). By substituting Equation (6.12) into Equation (6.5b) and using Equation (6.6), after some algebra the following is obtained:

∂ Bl ,m (τ ) + DRl(l + 1)Bl ,m (τ ) = 0 ∂τ

(6.13)

which has a general solution: Bl ,m (τ ) = Cl ,m exp[−l(l + 1)DRτ ]

(6.14)

Combining Equation (6.12) and Equation (6.14) gives the following expression for f(Ω ,τ): ∞

f (Ω , τ ) =

l

∑∑C

Y (Ω)exp[−l(l + 1)DRτ ]

l ,m l ,m

(6.15)

l = 0 m=− l

The constants Cl,m can be determined from the initial condition given in Equation (6.8) and the expansion of the δ-function in spherical harmonics, according to Equation (6.10c). This results in: Cl ,m = Yl∗,m (Ω 0 )

(6.16)

and the propagator becomes: ∞

P(Ω 0|Ω , τ ) =

l

∑∑Y

∗ l ,m

l= 0 m=− l

(Ω 0 )Yl ,m (Ω)exp[−l(l + 1)DRτ ]

(6.17)

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Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications

For the evaluation of G(τ), we now need to multiply Equation (6.17) with Y2,0(Ω 0)Y *2,0(Ω ) and to integrate over Ω 0 and Ω according to Equation (6.4). The fact that the spherical harmonics form an orthonormal set, as specified in Equation (6.10b), picks out one and only one term from the summation in Equation (6.17), that with l = 2, m = 0. Thus, we finally obtain: G2 (τ ) =

1 exp[−6DRτ ] 4π

(6.18)

This is identical to the exponential function we “guessed” in Section 2.3, if we identify: 1 6DR

τc =

(6.19)

The one-sided Fourier t